Theorem wfaxsep 45093

Description: The class of well-founded sets models the Axiom of Separation ax-sep 5236. 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 45080	. 2 ⊢ (∀𝑧 ∈ 𝑊 𝒫 𝑧 ⊆ 𝑊 → ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
2		pwwf 9706	. . . 4 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑧 ∈ ∪ (𝑅1 “ On))
3		r1elssi 9704	. . . 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 2823	. . 3 ⊢ (𝑧 ∈ 𝑊 ↔ 𝑧 ∈ ∪ (𝑅1 “ On))
7	5	sseq2i 3959	. . 3 ⊢ (𝒫 𝑧 ⊆ 𝑊 ↔ 𝒫 𝑧 ⊆ ∪ (𝑅1 “ On))
8	4, 6, 7	3imtr4i 292	. 2 ⊢ (𝑧 ∈ 𝑊 → 𝒫 𝑧 ⊆ 𝑊)
9	1, 8	mprg 3053	1 ⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4549  ∪ cuni 4858   “ cima 5622  Oncon0 6312  𝑅₁cr1 9661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-r1 9663  df-rank 9664

This theorem is referenced by: (None)