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Mirrors > Home > ILE Home > Th. List > zfpair2 | GIF version |
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4205. (Contributed by NM, 14-Nov-2006.) |
Ref | Expression |
---|---|
zfpair2 | ⊢ {𝑥, 𝑦} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pr 4205 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
2 | 1 | bm1.3ii 4121 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
3 | dfcleq 2171 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 2740 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 3612 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 227 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | 6 | albii 1470 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
8 | 3, 7 | bitri 184 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
9 | 8 | exbii 1605 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 2, 9 | mpbir 146 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
11 | 10 | issetri 2746 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∨ wo 708 ∀wal 1351 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 {cpr 3592 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 |
This theorem is referenced by: prexg 4207 onintexmid 4568 funopg 5245 |
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