Theorem wfaxsep 45732

Description: The class of well-founded sets models the Axiom of Separation ax-sep 5256. 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	⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝑦,𝑊

Allowed substitution hints:   𝜑(𝑥)   𝑊(𝑥, 𝑧)

Proof of Theorem wfaxsep

Step	Hyp	Ref	Expression
1		ssclaxsep 45719	. 2 ⊢ (∀𝑧 ∈ 𝑊 𝒫 𝑧 ⊆ 𝑊 → ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
2		pwwf 9777	. . . 4 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑧 ∈ ∪ (𝑅1 “ On))
3		r1elssi 9775	. . . 4 ⊢ (𝒫 𝑧 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
4	2, 3	sylbi 220	. . 3 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
5		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
6	5	eleq2i 2854	. . 3 ⊢ (𝑧 ∈ 𝑊 ↔ 𝑧 ∈ ∪ (𝑅1 “ On))
7	5	sseq2i 3965	. . 3 ⊢ (𝒫 𝑧 ⊆ 𝑊 ↔ 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
8	4, 6, 7	3imtr4i 295	. 2 ⊢ (𝑧 ∈ 𝑊 → 𝒫 𝑧 ⊆ 𝑊)
9	1, 8	mprg 3084	1 ⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871   “ cima 5663  Oncon0 6360  𝑅₁cr1 9732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-r1 9734  df-rank 9735

This theorem is used by: (None)