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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-zfpair2 | GIF version |
Description: Proof of zfpair2 4193 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-zfpair2 | ⊢ {𝑥, 𝑦} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-bdeq 13815 | . . . . 5 ⊢ BOUNDED 𝑤 = 𝑥 | |
2 | ax-bdeq 13815 | . . . . 5 ⊢ BOUNDED 𝑤 = 𝑦 | |
3 | 1, 2 | ax-bdor 13811 | . . . 4 ⊢ BOUNDED (𝑤 = 𝑥 ∨ 𝑤 = 𝑦) |
4 | ax-pr 4192 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
5 | 3, 4 | bdbm1.3ii 13886 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | dfcleq 2164 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
7 | vex 2733 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
8 | 7 | elpr 3602 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
9 | 8 | bibi2i 226 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 9 | albii 1463 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
11 | 6, 10 | bitri 183 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
12 | 11 | exbii 1598 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
13 | 5, 12 | mpbir 145 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
14 | 13 | issetri 2739 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 703 ∀wal 1346 = wceq 1348 ∃wex 1485 ∈ wcel 2141 Vcvv 2730 {cpr 3582 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-pr 4192 ax-bdor 13811 ax-bdeq 13815 ax-bdsep 13879 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 |
This theorem is referenced by: bj-prexg 13906 |
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