Theorem pltval3 18331

Description: Alternate expression for the "less than" relation. (dfpss3 4084 analog.) (Contributed by NM, 4-Nov-2011.)

Hypotheses

Ref	Expression
pleval2.b	⊢ 𝐵 = (Base‘𝐾)
pleval2.l	⊢ ≤ = (le‘𝐾)
pleval2.s	⊢ < = (lt‘𝐾)

Assertion

Ref	Expression
pltval3	⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))

Proof of Theorem pltval3

Step	Hyp	Ref	Expression
1		pleval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		pleval2.s	. . 3 ⊢ < = (lt‘𝐾)
3	1, 2	pltval 18324	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
4		pleval2.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	4, 1	posref 18310	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
6	5	3adant3 1130	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ 𝑋)
7		breq1 5151	. . . . . . 7 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
8	6, 7	syl5ibcom 244	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 → 𝑌 ≤ 𝑋))
9	8	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 = 𝑌 → 𝑌 ≤ 𝑋))
10	4, 1	posasymb 18311	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
11	10	biimpd 228	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))
12	11	expdimp 452	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑌 ≤ 𝑋 → 𝑋 = 𝑌))
13	9, 12	impbid 211	. . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 = 𝑌 ↔ 𝑌 ≤ 𝑋))
14	13	necon3abid 2974	. . 3 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ≠ 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))
15	14	pm5.32da 578	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
16	3, 15	bitrd 279	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937   class class class wbr 5148  ‘cfv 6548  Basecbs 17180  lecple 17240  Posetcpo 18299  ltcplt 18300

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-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  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-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-proset 18287  df-poset 18305  df-plt 18322

This theorem is referenced by:  tltnle  18414  opltcon3b  38676

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