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Theorem eldju1st 9946

Description: The first component of an element of a disjoint union is either ∅ or 1_o. (Contributed by AV, 26-Jun-2022.)

**Assertion**
Ref	Expression
eldju1st	⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1^st ‘𝑋) = ∅ ∨ (1^st ‘𝑋) = 1_o))

**Proof of Theorem eldju1st**
Step	Hyp	Ref	Expression
1		djuss 9943	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1_o} × (𝐴 ∪ 𝐵))
2		ssel2 3972	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1_o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1_o} × (𝐴 ∪ 𝐵)))
3		xp1st 8024	. . 3 ⊢ (𝑋 ∈ ({∅, 1_o} × (𝐴 ∪ 𝐵)) → (1^st ‘𝑋) ∈ {∅, 1_o})
4		elpri 4652	. . 3 ⊢ ((1^st ‘𝑋) ∈ {∅, 1_o} → ((1^st ‘𝑋) = ∅ ∨ (1^st ‘𝑋) = 1_o))
5	2, 3, 4	3syl 18	. 2 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1_o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → ((1^st ‘𝑋) = ∅ ∨ (1^st ‘𝑋) = 1_o))
6	1, 5	mpan 688	1 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1^st ‘𝑋) = ∅ ∨ (1^st ‘𝑋) = 1_o))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∪ cun 3943 ⊆ wss 3945 ∅c0 4323 {cpr 4631 × cxp 5675 ‘cfv 6547 1^st c1st 7990 1_oc1o 8478 ⊔ cdju 9921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pr 5428 ax-un 7739

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-suc 6375 df-iota 6499 df-fun 6549 df-fv 6555 df-1st 7992 df-2nd 7993 df-1o 8485 df-dju 9924 df-inl 9925 df-inr 9926

This theorem is referenced by: (None)

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