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	⊢ ∀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 44975	. 2 ⊢ (∀𝑧 ∈ 𝑊 𝒫 𝑧 ⊆ 𝑊 → ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
2		pwwf 9843	. . . 4 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑧 ∈ ∪ (𝑅1 “ On))
3		r1elssi 9841	. . . 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 2832	. . 3 ⊢ (𝑧 ∈ 𝑊 ↔ 𝑧 ∈ ∪ (𝑅1 “ On))
7	5	sseq2i 4012	. . 3 ⊢ (𝒫 𝑧 ⊆ 𝑊 ↔ 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
8	4, 6, 7	3imtr4i 292	. 2 ⊢ (𝑧 ∈ 𝑊 → 𝒫 𝑧 ⊆ 𝑊)
9	1, 8	mprg 3066	1 ⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

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)