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Theorem eldju1st 9948

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 9945	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1_o} × (𝐴 ∪ 𝐵))
2		ssel2 3971	. . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1_o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1_o} × (𝐴 ∪ 𝐵)))
3		xp1st 8026	. . 3 ⊢ (𝑋 ∈ ({∅, 1_o} × (𝐴 ∪ 𝐵)) → (1^st ‘𝑋) ∈ {∅, 1_o})
4		elpri 4653	. . 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 3942 ⊆ wss 3944 ∅c0 4322 {cpr 4632 × cxp 5676 ‘cfv 6549 1^st c1st 7992 1_oc1o 8480 ⊔ cdju 9923

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 5300 ax-nul 5307 ax-pr 5429 ax-un 7741

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 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-suc 6377 df-iota 6501 df-fun 6551 df-fv 6557 df-1st 7994 df-2nd 7995 df-1o 8487 df-dju 9926 df-inl 9927 df-inr 9928

This theorem is referenced by: (None)

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