Theorem wfaxsep 45314

Description: The class of well-founded sets models the Axiom of Separation ax-sep 5242. 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 45301	. 2 ⊢ (∀𝑧 ∈ 𝑊 𝒫 𝑧 ⊆ 𝑊 → ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
2		pwwf 9724	. . . 4 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑧 ∈ ∪ (𝑅1 “ On))
3		r1elssi 9722	. . . 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 3964	. . 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 3061   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864   “ cima 5628  Oncon0 6318  𝑅₁cr1 9679

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 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

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 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-r1 9681  df-rank 9682

This theorem is referenced by: (None)