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Theorem wfaxsep 44988
Description: The class of well-founded sets models the Axiom of Separation ax-sep 5294. Actually, our statement is stronger, since it is an instance of Separation only when all quantifiers in 𝜑 are relativized to 𝑊. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
Hypothesis
Ref Expression
wfax.1 𝑊 = (𝑅1 “ On)
Assertion
Ref Expression
wfaxsep 𝑧𝑊𝑦𝑊𝑥𝑊 (𝑥𝑦 ↔ (𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝑦,𝑊
Allowed substitution hints:   𝜑(𝑥)   𝑊(𝑥,𝑧)

Proof of Theorem wfaxsep
StepHypRef Expression
1 ssclaxsep 44975 . 2 (∀𝑧𝑊 𝒫 𝑧𝑊 → ∀𝑧𝑊𝑦𝑊𝑥𝑊 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
2 pwwf 9843 . . . 4 (𝑧 (𝑅1 “ On) ↔ 𝒫 𝑧 (𝑅1 “ On))
3 r1elssi 9841 . . . 4 (𝒫 𝑧 (𝑅1 “ On) → 𝒫 𝑧 (𝑅1 “ On))
42, 3sylbi 217 . . 3 (𝑧 (𝑅1 “ On) → 𝒫 𝑧 (𝑅1 “ On))
5 wfax.1 . . . 4 𝑊 = (𝑅1 “ On)
65eleq2i 2832 . . 3 (𝑧𝑊𝑧 (𝑅1 “ On))
75sseq2i 4012 . . 3 (𝒫 𝑧𝑊 ↔ 𝒫 𝑧 (𝑅1 “ On))
84, 6, 73imtr4i 292 . 2 (𝑧𝑊 → 𝒫 𝑧𝑊)
91, 8mprg 3066 1 𝑧𝑊𝑦𝑊𝑥𝑊 (𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3060  wrex 3069  wss 3950  𝒫 cpw 4598   cuni 4905  cima 5686  Oncon0 6382  𝑅1cr1 9798
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-om 7884  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-r1 9800  df-rank 9801
This theorem is referenced by: (None)
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