Theorem wfaxsep 45425

Description: The class of well-founded sets models the Axiom of Separation ax-sep 5231. 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 45412	. 2 ⊢ (∀𝑧 ∈ 𝑊 𝒫 𝑧 ⊆ 𝑊 → ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
2		pwwf 9720	. . . 4 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑧 ∈ ∪ (𝑅1 “ On))
3		r1elssi 9718	. . . 4 ⊢ (𝒫 𝑧 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
4	2, 3	sylbi 217	. . 3 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
5		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
6	5	eleq2i 2829	. . 3 ⊢ (𝑧 ∈ 𝑊 ↔ 𝑧 ∈ ∪ (𝑅1 “ On))
7	5	sseq2i 3952	. . 3 ⊢ (𝒫 𝑧 ⊆ 𝑊 ↔ 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
8	4, 6, 7	3imtr4i 292	. 2 ⊢ (𝑧 ∈ 𝑊 → 𝒫 𝑧 ⊆ 𝑊)
9	1, 8	mprg 3058	1 ⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   “ cima 5625  Oncon0 6315  𝑅₁cr1 9675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-r1 9677  df-rank 9678

This theorem is referenced by: (None)