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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 5433. See zfpair 5425 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 5433 | . . . 4 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | |
2 | 1 | bm1.3ii 5306 | . . 3 ⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
3 | dfcleq 2721 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 3477 | . . . . . . . 8 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 4656 | . . . . . . 7 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 336 | . . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | 6 | albii 1813 | . . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
8 | 3, 7 | bitri 274 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
9 | 8 | exbii 1842 | . . 3 ⊢ (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
10 | 2, 9 | mpbir 230 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
11 | 10 | issetri 3490 | 1 ⊢ {𝑥, 𝑦} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 845 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3473 {cpr 4634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-un 3954 df-sn 4633 df-pr 4635 |
This theorem is referenced by: vsnex 5435 prex 5438 pwssun 5577 xpsspw 5815 funopg 6592 fiint 9356 brdom7disj 10562 brdom6disj 10563 2pthfrgrrn 30112 sprval 46848 prprval 46883 reupr 46891 uspgrimprop 47249 |
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