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| Mirrors > Home > MPE Home > Th. List > Mathboxes > permaxext | Structured version Visualization version GIF version | ||
| Description: The Axiom of Extensionality ax-ext 2734 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.) |
| Ref | Expression |
|---|---|
| permmodel.1 | ⊢ 𝐹:V–1-1-onto→V |
| permmodel.2 | ⊢ 𝑅 = (◡𝐹 ∘ E ) |
| Ref | Expression |
|---|---|
| permaxext | ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | permmodel.1 | . . . . . 6 ⊢ 𝐹:V–1-1-onto→V | |
| 2 | permmodel.2 | . . . . . 6 ⊢ 𝑅 = (◡𝐹 ∘ E ) | |
| 3 | vex 3458 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 4 | vex 3458 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 5 | 1, 2, 3, 4 | brpermmodel 45576 | . . . . 5 ⊢ (𝑧𝑅𝑥 ↔ 𝑧 ∈ (𝐹‘𝑥)) |
| 6 | vex 3458 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 7 | 1, 2, 3, 6 | brpermmodel 45576 | . . . . 5 ⊢ (𝑧𝑅𝑦 ↔ 𝑧 ∈ (𝐹‘𝑦)) |
| 8 | 5, 7 | bibi12i 341 | . . . 4 ⊢ ((𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦))) |
| 9 | 8 | albii 1839 | . . 3 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦))) |
| 10 | dfcleq 2755 | . . 3 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑧(𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ (𝐹‘𝑦))) | |
| 11 | 9, 10 | bitr4i 280 | . 2 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)) |
| 12 | f1of1 6805 | . . . . 5 ⊢ (𝐹:V–1-1-onto→V → 𝐹:V–1-1→V) | |
| 13 | 1, 12 | ax-mp 5 | . . . 4 ⊢ 𝐹:V–1-1→V |
| 14 | f1veqaeq 7240 | . . . 4 ⊢ ((𝐹:V–1-1→V ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) | |
| 15 | 13, 14 | mpan 700 | . . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) |
| 16 | 15 | el2v 3461 | . 2 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) |
| 17 | 11, 16 | sylbi 219 | 1 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∀wal 1558 = wceq 1560 ∈ wcel 2142 Vcvv 3454 class class class wbr 5100 E cep 5546 ◡ccnv 5646 ∘ ccom 5651 –1-1→wf1 6518 –1-1-onto→wf1o 6520 ‘cfv 6521 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5542 df-eprel 5547 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-f1o 6528 df-fv 6529 |
| This theorem is referenced by: nregmodelaxext 45591 |
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