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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 4091. (Contributed by NM, 14-Nov-2006.) |
Ref | Expression |
---|---|
zfpair2 | ⊢ {𝑥, 𝑦} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pr 4091 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
2 | 1 | bm1.3ii 4009 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
3 | dfcleq 2109 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 2660 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 3514 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 226 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | 6 | albii 1429 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
8 | 3, 7 | bitri 183 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
9 | 8 | exbii 1567 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 2, 9 | mpbir 145 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
11 | 10 | issetri 2666 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 680 ∀wal 1312 = wceq 1314 ∃wex 1451 ∈ wcel 1463 Vcvv 2657 {cpr 3494 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 |
This theorem is referenced by: prexg 4093 onintexmid 4447 funopg 5115 |
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