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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 5432. See zfpair 5421 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 5432 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
| 2 | 1 | sepexi 5301 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
| 3 | dfcleq 2730 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
| 4 | vex 3484 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
| 5 | 4 | elpr 4650 | . . . . . . 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 1819 | . . . . 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 1848 | . . 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 3499 | 1 ⊢ {𝑥, 𝑦} ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∨ wo 848 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: vsnex 5434 prex 5437 pwssun 5575 xpsspw 5819 funopg 6600 fiint 9366 fiintOLD 9367 brdom7disj 10571 brdom6disj 10572 2pthfrgrrn 30301 sprval 47466 prprval 47501 reupr 47509 uspgrimprop 47873 |
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