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Mirrors > Home > MPE Home > Th. List > eldju2ndr | Structured version Visualization version GIF version |
Description: The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.) |
Ref | Expression |
---|---|
eldju2ndr | ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9939 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 1 | eleq2i 2831 | . . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
3 | elun 4163 | . . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) | |
4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) |
5 | elxp6 8047 | . . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴))) | |
6 | elsni 4648 | . . . . . . 7 ⊢ ((1st ‘𝑋) ∈ {∅} → (1st ‘𝑋) = ∅) | |
7 | eqneqall 2949 | . . . . . . 7 ⊢ ((1st ‘𝑋) = ∅ → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((1st ‘𝑋) ∈ {∅} → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
9 | 8 | ad2antrl 728 | . . . . 5 ⊢ ((𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
10 | 5, 9 | sylbi 217 | . . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
11 | elxp6 8047 | . . . . 5 ⊢ (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵))) | |
12 | simprr 773 | . . . . . 6 ⊢ ((𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵)) → (2nd ‘𝑋) ∈ 𝐵) | |
13 | 12 | a1d 25 | . . . . 5 ⊢ ((𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
14 | 11, 13 | sylbi 217 | . . . 4 ⊢ (𝑋 ∈ ({1o} × 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
15 | 10, 14 | jaoi 857 | . . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
16 | 4, 15 | sylbi 217 | . 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
17 | 16 | imp 406 | 1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∪ cun 3961 ∅c0 4339 {csn 4631 〈cop 4637 × cxp 5687 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 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-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 df-dju 9939 |
This theorem is referenced by: updjudhf 9969 |
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