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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 9323 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 1 | eleq2i 2903 | . . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
3 | elun 4118 | . . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) |
5 | elxp6 7716 | . . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴))) | |
6 | elsni 4577 | . . . . . . 7 ⊢ ((1st ‘𝑋) ∈ {∅} → (1st ‘𝑋) = ∅) | |
7 | eqneqall 3026 | . . . . . . 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 219 | . . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
11 | elxp6 7716 | . . . . 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 219 | . . . 4 ⊢ (𝑋 ∈ ({1o} × 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
15 | 10, 14 | jaoi 853 | . . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
16 | 4, 15 | sylbi 219 | . 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
17 | 16 | imp 409 | 1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∪ cun 3927 ∅c0 4284 {csn 4560 〈cop 4566 × cxp 5546 ‘cfv 6348 1st c1st 7680 2nd c2nd 7681 1oc1o 8088 ⊔ cdju 9320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7682 df-2nd 7683 df-dju 9323 |
This theorem is referenced by: updjudhf 9353 |
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