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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 9941 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
| 2 | 1 | eleq2i 2833 | . . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | 
| 3 | elun 4153 | . . . 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 8048 | . . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴))) | |
| 6 | elsni 4643 | . . . . . . 7 ⊢ ((1st ‘𝑋) ∈ {∅} → (1st ‘𝑋) = ∅) | |
| 7 | eqneqall 2951 | . . . . . . 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 8048 | . . . . 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 858 | . . 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 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∪ cun 3949 ∅c0 4333 {csn 4626 〈cop 4632 × cxp 5683 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 1oc1o 8499 ⊔ cdju 9938 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 df-dju 9941 | 
| This theorem is referenced by: updjudhf 9971 | 
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