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Theorem wfaxsep 44950
Description: The class of well-founded sets models the Axiom of Separation ax-sep 5301. 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 44946 . 2 (∀𝑧𝑊 𝒫 𝑧𝑊 → ∀𝑧𝑊𝑦𝑊𝑥𝑊 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
2 pwwf 9844 . . . 4 (𝑧 (𝑅1 “ On) ↔ 𝒫 𝑧 (𝑅1 “ On))
3 r1elssi 9842 . . . 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 2830 . . 3 (𝑧𝑊𝑧 (𝑅1 “ On))
75sseq2i 4024 . . 3 (𝒫 𝑧𝑊 ↔ 𝒫 𝑧 (𝑅1 “ On))
84, 6, 73imtr4i 292 . 2 (𝑧𝑊 → 𝒫 𝑧𝑊)
91, 8mprg 3064 1 𝑧𝑊𝑦𝑊𝑥𝑊 (𝑥𝑦 ↔ (𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  wss 3962  𝒫 cpw 4604   cuni 4911  cima 5691  Oncon0 6385  𝑅1cr1 9799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-r1 9801  df-rank 9802
This theorem is referenced by: (None)
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