hash1snb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hash1snb		Structured version Visualization version GIF version

Theorem hash1snb 14410

Description: The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.)

**Assertion**
Ref	Expression
hash1snb	⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎}))

Distinct variable group: 𝑉,𝑎 Allowed substitution hint: 𝑊(𝑎)

**Proof of Theorem hash1snb**
Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . 9 ⊢ ((♯‘𝑉) = 1 → (♯‘𝑉) = 1)
2		hash1 14395	. . . . . . . . 9 ⊢ (♯‘1_o) = 1
3	1, 2	eqtr4di 2786	. . . . . . . 8 ⊢ ((♯‘𝑉) = 1 → (♯‘𝑉) = (♯‘1_o))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → (♯‘𝑉) = (♯‘1_o))
5		1onn 8660	. . . . . . . . 9 ⊢ 1_o ∈ ω
6		nnfi 9191	. . . . . . . . 9 ⊢ (1_o ∈ ω → 1_o ∈ Fin)
7	5, 6	mp1i 13	. . . . . . . 8 ⊢ ((♯‘𝑉) = 1 → 1_o ∈ Fin)
8		hashen 14338	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 1_o ∈ Fin) → ((♯‘𝑉) = (♯‘1_o) ↔ 𝑉 ≈ 1_o))
9	7, 8	sylan2 592	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → ((♯‘𝑉) = (♯‘1_o) ↔ 𝑉 ≈ 1_o))
10	4, 9	mpbid 231	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → 𝑉 ≈ 1_o)
11		en1 9045	. . . . . 6 ⊢ (𝑉 ≈ 1_o ↔ ∃𝑎 𝑉 = {𝑎})
12	10, 11	sylib 217	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → ∃𝑎 𝑉 = {𝑎})
13	12	ex 412	. . . 4 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎}))
14	13	a1d 25	. . 3 ⊢ (𝑉 ∈ Fin → (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎})))
15		hashinf 14326	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ 𝑉 ∈ Fin) → (♯‘𝑉) = +∞)
16		eqeq1 2732	. . . . . 6 ⊢ ((♯‘𝑉) = +∞ → ((♯‘𝑉) = 1 ↔ +∞ = 1))
17		1re 11244	. . . . . . . 8 ⊢ 1 ∈ ℝ
18		renepnf 11292	. . . . . . . 8 ⊢ (1 ∈ ℝ → 1 ≠ +∞)
19		df-ne 2938	. . . . . . . . 9 ⊢ (1 ≠ +∞ ↔ ¬ 1 = +∞)
20		pm2.21 123	. . . . . . . . 9 ⊢ (¬ 1 = +∞ → (1 = +∞ → ∃𝑎 𝑉 = {𝑎}))
21	19, 20	sylbi 216	. . . . . . . 8 ⊢ (1 ≠ +∞ → (1 = +∞ → ∃𝑎 𝑉 = {𝑎}))
22	17, 18, 21	mp2b 10	. . . . . . 7 ⊢ (1 = +∞ → ∃𝑎 𝑉 = {𝑎})
23	22	eqcoms 2736	. . . . . 6 ⊢ (+∞ = 1 → ∃𝑎 𝑉 = {𝑎})
24	16, 23	biimtrdi 252	. . . . 5 ⊢ ((♯‘𝑉) = +∞ → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎}))
25	15, 24	syl 17	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ 𝑉 ∈ Fin) → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎}))
26	25	expcom 413	. . 3 ⊢ (¬ 𝑉 ∈ Fin → (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎})))
27	14, 26	pm2.61i 182	. 2 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎}))
28		fveq2 6897	. . . 4 ⊢ (𝑉 = {𝑎} → (♯‘𝑉) = (♯‘{𝑎}))
29		hashsng 14360	. . . . 5 ⊢ (𝑎 ∈ V → (♯‘{𝑎}) = 1)
30	29	elv 3477	. . . 4 ⊢ (♯‘{𝑎}) = 1
31	28, 30	eqtrdi 2784	. . 3 ⊢ (𝑉 = {𝑎} → (♯‘𝑉) = 1)
32	31	exlimiv 1926	. 2 ⊢ (∃𝑎 𝑉 = {𝑎} → (♯‘𝑉) = 1)
33	27, 32	impbid1 224	1 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 {csn 4629 class class class wbr 5148 ‘cfv 6548 ωcom 7870 1_oc1o 8479 ≈ cen 8960 Fincfn 8963 ℝcr 11137 1c1 11139 +∞cpnf 11275 ♯chash 14321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-hash 14322

This theorem is referenced by: hash1n0 14412 hashle2pr 14470 hashge2el2difr 14474 hash1to3 14484 cshwrepswhash1 17071 symgvalstruct 19350 symgvalstructOLD 19351 c0snmgmhm 20400 mat1scmat 22440 tgldim0eq 28306 lfuhgr1v0e 29066 usgr1v0e 29138 nbgr1vtx 29170 uvtx01vtx 29209 cplgr1vlem 29241 cplgr1v 29242 1loopgrvd2 29316 vdgn1frgrv2 30105 frgrwopreg1 30127 frgrwopreg2 30128 extdg1id 33351

W3C validator

	
		OSZAR »