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| Mirrors > Home > MPE Home > Th. List > Mathboxes > permaxsep | Structured version Visualization version GIF version | ||
| Description: The Axiom of Separation
ax-sep 5246 holds in permutation models. Part of
Exercise II.9.2 of [Kunen2] p. 148.
Note that, to prove that an instance of Separation holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Separation holds. (Contributed by Eric Schmidt, 6-Nov-2025.) |
| Ref | Expression |
|---|---|
| permmodel.1 | ⊢ 𝐹:V–1-1-onto→V |
| permmodel.2 | ⊢ 𝑅 = (◡𝐹 ∘ E ) |
| Ref | Expression |
|---|---|
| permaxsep | ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvex 6880 | . 2 ⊢ (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ∈ V | |
| 2 | nfcv 2924 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 | |
| 3 | nfrab1 3434 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} | |
| 4 | 2, 3 | nffv 6877 | . . . 4 ⊢ Ⅎ𝑥(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) |
| 5 | 4 | nfeq2 2941 | . . 3 ⊢ Ⅎ𝑥 𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) |
| 6 | breq2 5104 | . . . 4 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → (𝑥𝑅𝑦 ↔ 𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}))) | |
| 7 | 6 | bibi1d 345 | . . 3 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → ((𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) ↔ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑)))) |
| 8 | 5, 7 | albid 2257 | . 2 ⊢ (𝑦 = (◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) → (∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) ↔ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑)))) |
| 9 | permmodel.1 | . . . . 5 ⊢ 𝐹:V–1-1-onto→V | |
| 10 | permmodel.2 | . . . . 5 ⊢ 𝑅 = (◡𝐹 ∘ E ) | |
| 11 | vex 3458 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 12 | fvex 6880 | . . . . . 6 ⊢ (𝐹‘𝑧) ∈ V | |
| 13 | 12 | rabex 5295 | . . . . 5 ⊢ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ∈ V |
| 14 | 9, 10, 11, 13 | brpermmodelcnv 45577 | . . . 4 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) |
| 15 | rabid 3435 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥 ∈ (𝐹‘𝑧) ∧ 𝜑)) | |
| 16 | vex 3458 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 17 | 9, 10, 11, 16 | brpermmodel 45576 | . . . . . 6 ⊢ (𝑥𝑅𝑧 ↔ 𝑥 ∈ (𝐹‘𝑧)) |
| 18 | 17 | bicomi 226 | . . . . 5 ⊢ (𝑥 ∈ (𝐹‘𝑧) ↔ 𝑥𝑅𝑧) |
| 19 | 15, 18 | bianbi 636 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑} ↔ (𝑥𝑅𝑧 ∧ 𝜑)) |
| 20 | 14, 19 | bitri 277 | . . 3 ⊢ (𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑)) |
| 21 | 20 | ax-gen 1815 | . 2 ⊢ ∀𝑥(𝑥𝑅(◡𝐹‘{𝑥 ∈ (𝐹‘𝑧) ∣ 𝜑}) ↔ (𝑥𝑅𝑧 ∧ 𝜑)) |
| 22 | 1, 8, 21 | ceqsexv2d 3503 | 1 ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 208 ∧ wa 399 ∀wal 1558 = wceq 1560 ∃wex 1799 ∈ wcel 2142 {crab 3414 Vcvv 3454 class class class wbr 5100 E cep 5546 ◡ccnv 5646 ∘ ccom 5651 –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-nfc 2911 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-pw 4557 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-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 |
| This theorem is referenced by: (None) |
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