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Mirrors > Home > MPE Home > Th. List > zfpair2 | Structured version Visualization version GIF version |
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 5438. See zfpair 5427 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.) |
Ref | Expression |
---|---|
zfpair2 | ⊢ {𝑥, 𝑦} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pr 5438 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
2 | 1 | sepexi 5307 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
3 | dfcleq 2728 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 3482 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 4655 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 337 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | 6 | albii 1816 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
8 | 3, 7 | bitri 275 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
9 | 8 | exbii 1845 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 2, 9 | mpbir 231 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
11 | 10 | issetri 3497 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 {cpr 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: vsnex 5440 prex 5443 pwssun 5580 xpsspw 5822 funopg 6602 fiint 9364 fiintOLD 9365 brdom7disj 10569 brdom6disj 10570 2pthfrgrrn 30311 sprval 47404 prprval 47439 reupr 47447 uspgrimprop 47811 |
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