Theorem wfaxpow 45180

Description: The class of well-founded sets models the Axioms of Power Sets. Part of Corollary II.2.9 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)

Hypothesis

Ref	Expression
wfax.1	⊢ 𝑊 = ∪ (𝑅1 “ On)

Assertion

Ref	Expression
wfaxpow	⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑊

Proof of Theorem wfaxpow

Step	Hyp	Ref	Expression
1		trwf 45142	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . . 5 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5210	. . . . 5 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On)))
4	2, 3	ax-mp 5	. . . 4 ⊢ (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On))
5	1, 4	mpbir 231	. . 3 ⊢ Tr 𝑊
6		pwclaxpow 45167	. . 3 ⊢ ((Tr 𝑊 ∧ ∀𝑥 ∈ 𝑊 (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊) → ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
7	5, 6	mpan 690	. 2 ⊢ (∀𝑥 ∈ 𝑊 (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 → ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8		pwwf 9717	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 216	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
10		r1elssi 9715	. . . . 5 ⊢ (𝒫 𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ⊆ ∪ (𝑅1 “ On))
11		dfss2 3917	. . . . . 6 ⊢ (𝒫 𝑥 ⊆ ∪ (𝑅1 “ On) ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥)
12		eleq1 2822	. . . . . 6 ⊢ ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥 → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
13	11, 12	sylbi 217	. . . . 5 ⊢ (𝒫 𝑥 ⊆ ∪ (𝑅1 “ On) → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
14	9, 10, 13	3syl 18	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
15	9, 14	mpbird 257	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On))
16	2	eleq2i 2826	. . 3 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
17	2	ineq2i 4167	. . . 4 ⊢ (𝒫 𝑥 ∩ 𝑊) = (𝒫 𝑥 ∩ ∪ (𝑅1 “ On))
18	17, 2	eleq12i 2827	. . 3 ⊢ ((𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On))
19	15, 16, 18	3imtr4i 292	. 2 ⊢ (𝑥 ∈ 𝑊 → (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊)
20	7, 19	mprg 3055	1 ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4552  ∪ cuni 4861  Tr wtr 5203   “ cima 5625  Oncon0 6315  𝑅₁cr1 9672

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  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 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9674  df-rank 9675

This theorem is referenced by: (None)