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Mirrors > Home > MPE Home > Th. List > eldju1st | Structured version Visualization version GIF version |
Description: The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.) |
Ref | Expression |
---|---|
eldju1st | ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuss 9958 | . 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) | |
2 | ssel2 3990 | . . 3 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵))) | |
3 | xp1st 8045 | . . 3 ⊢ (𝑋 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → (1st ‘𝑋) ∈ {∅, 1o}) | |
4 | elpri 4654 | . . 3 ⊢ ((1st ‘𝑋) ∈ {∅, 1o} → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o)) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (((𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) ∧ 𝑋 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o)) |
6 | 1, 5 | mpan 690 | 1 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 {cpr 4633 × cxp 5687 ‘cfv 6563 1st c1st 8011 1oc1o 8498 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 df-1o 8505 df-dju 9939 df-inl 9940 df-inr 9941 |
This theorem is referenced by: (None) |
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