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Mirrors > Home > ILE Home > Th. List > eldju2ndr | 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 6891 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 1 | eleq2i 2184 | . . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
3 | elun 3187 | . . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) | |
4 | 2, 3 | bitri 183 | . . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) |
5 | elxp6 6035 | . . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴))) | |
6 | elsni 3515 | . . . . . . 7 ⊢ ((1st ‘𝑋) ∈ {∅} → (1st ‘𝑋) = ∅) | |
7 | eqneqall 2295 | . . . . . . 7 ⊢ ((1st ‘𝑋) = ∅ → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) | |
8 | 6, 7 | syl 14 | . . . . . 6 ⊢ ((1st ‘𝑋) ∈ {∅} → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
9 | 8 | ad2antrl 481 | . . . . 5 ⊢ ((𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
10 | 5, 9 | sylbi 120 | . . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
11 | elxp6 6035 | . . . . 5 ⊢ (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵))) | |
12 | simprr 506 | . . . . . 6 ⊢ ((𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵)) → (2nd ‘𝑋) ∈ 𝐵) | |
13 | 12 | a1d 22 | . . . . 5 ⊢ ((𝑋 = 〈(1st ‘𝑋), (2nd ‘𝑋)〉 ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
14 | 11, 13 | sylbi 120 | . . . 4 ⊢ (𝑋 ∈ ({1o} × 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
15 | 10, 14 | jaoi 690 | . . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
16 | 4, 15 | sylbi 120 | . 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐵)) |
17 | 16 | imp 123 | 1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 682 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ∪ cun 3039 ∅c0 3333 {csn 3497 〈cop 3500 × cxp 4507 ‘cfv 5093 1st c1st 6004 2nd c2nd 6005 1oc1o 6274 ⊔ cdju 6890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-1st 6006 df-2nd 6007 df-dju 6891 |
This theorem is referenced by: updjudhf 6932 |
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