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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 9924 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 1 | eleq2i 2817 | . . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
3 | elun 4141 | . . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) | |
4 | 2, 3 | bitri 274 | . . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) |
5 | elxp6 8025 | . . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴))) | |
6 | elsni 4641 | . . . . . . 7 ⊢ ((1st ‘𝑋) ∈ {∅} → (1st ‘𝑋) = ∅) | |
7 | eqneqall 2941 | . . . . . . 7 ⊢ ((1st ‘𝑋) = ∅ → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((1st ‘𝑋) ∈ {∅} → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
9 | 8 | ad2antrl 726 | . . . . 5 ⊢ ((𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
10 | 5, 9 | sylbi 216 | . . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
11 | elxp6 8025 | . . . . 5 ⊢ (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵))) | |
12 | simprr 771 | . . . . . 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 216 | . . . 4 ⊢ (𝑋 ∈ ({1o} × 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
15 | 10, 14 | jaoi 855 | . . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
16 | 4, 15 | sylbi 216 | . 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
17 | 16 | imp 405 | 1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∪ cun 3937 ∅c0 4318 {csn 4624 ⟨cop 4630 × cxp 5670 ‘cfv 6543 1st c1st 7989 2nd c2nd 7990 1oc1o 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 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
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 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7991 df-2nd 7992 df-dju 9924 |
This theorem is referenced by: updjudhf 9954 |
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