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Mirrors > Home > ILE Home > Th. List > eldju2ndl | GIF version |
Description: The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.) |
Ref | Expression |
---|---|
eldju2ndl | ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 7056 | . . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 1 | eleq2i 2256 | . . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
3 | elun 3291 | . . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) | |
4 | 2, 3 | bitri 184 | . . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵))) |
5 | elxp6 6188 | . . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴))) | |
6 | simprr 531 | . . . . . 6 ⊢ ((𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴)) → (2nd ‘𝑋) ∈ 𝐴) | |
7 | 6 | a1d 22 | . . . . 5 ⊢ ((𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {∅} ∧ (2nd ‘𝑋) ∈ 𝐴)) → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
8 | 5, 7 | sylbi 121 | . . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
9 | elxp6 6188 | . . . . 5 ⊢ (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵))) | |
10 | elsni 3625 | . . . . . . 7 ⊢ ((1st ‘𝑋) ∈ {1o} → (1st ‘𝑋) = 1o) | |
11 | 1n0 6451 | . . . . . . . 8 ⊢ 1o ≠ ∅ | |
12 | neeq1 2373 | . . . . . . . 8 ⊢ ((1st ‘𝑋) = 1o → ((1st ‘𝑋) ≠ ∅ ↔ 1o ≠ ∅)) | |
13 | 11, 12 | mpbiri 168 | . . . . . . 7 ⊢ ((1st ‘𝑋) = 1o → (1st ‘𝑋) ≠ ∅) |
14 | eqneqall 2370 | . . . . . . . 8 ⊢ ((1st ‘𝑋) = ∅ → ((1st ‘𝑋) ≠ ∅ → (2nd ‘𝑋) ∈ 𝐴)) | |
15 | 14 | com12 30 | . . . . . . 7 ⊢ ((1st ‘𝑋) ≠ ∅ → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
16 | 10, 13, 15 | 3syl 17 | . . . . . 6 ⊢ ((1st ‘𝑋) ∈ {1o} → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
17 | 16 | ad2antrl 490 | . . . . 5 ⊢ ((𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩ ∧ ((1st ‘𝑋) ∈ {1o} ∧ (2nd ‘𝑋) ∈ 𝐵)) → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
18 | 9, 17 | sylbi 121 | . . . 4 ⊢ (𝑋 ∈ ({1o} × 𝐵) → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
19 | 8, 18 | jaoi 717 | . . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
20 | 4, 19 | sylbi 121 | . 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ → (2nd ‘𝑋) ∈ 𝐴)) |
21 | 20 | imp 124 | 1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∪ cun 3142 ∅c0 3437 {csn 3607 ⟨cop 3610 × cxp 4639 ‘cfv 5231 1st c1st 6157 2nd c2nd 6158 1oc1o 6428 ⊔ cdju 7055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-suc 4386 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-iota 5193 df-fun 5233 df-fv 5239 df-1st 6159 df-2nd 6160 df-1o 6435 df-dju 7056 |
This theorem is referenced by: updjudhf 7097 subctctexmid 15154 |
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