Theorem wfaxpow 45454

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 45416	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . . 5 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5188	. . . . 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 233	. . 3 ⊢ Tr 𝑊
6		pwclaxpow 45441	. . 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 697	. 2 ⊢ (∀𝑥 ∈ 𝑊 (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 → ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8		pwwf 9726	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 218	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
10		r1elssi 9724	. . . . 5 ⊢ (𝒫 𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ⊆ ∪ (𝑅1 “ On))
11		dfss2 3902	. . . . . 6 ⊢ (𝒫 𝑥 ⊆ ∪ (𝑅1 “ On) ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥)
12		eleq1 2829	. . . . . 6 ⊢ ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥 → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
13	11, 12	sylbi 219	. . . . 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 259	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On))
16	2	eleq2i 2833	. . 3 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
17	2	ineq2i 4148	. . . 4 ⊢ (𝒫 𝑥 ∩ 𝑊) = (𝒫 𝑥 ∩ ∪ (𝑅1 “ On))
18	17, 2	eleq12i 2834	. . 3 ⊢ ((𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On))
19	15, 16, 18	3imtr4i 294	. 2 ⊢ (𝑥 ∈ 𝑊 → (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊)
20	7, 19	mprg 3061	1 ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ∩ cin 3883   ⊆ wss 3884  𝒫 cpw 4531  ∪ cuni 4840  Tr wtr 5181   “ cima 5623  Oncon0 6313  𝑅₁cr1 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-r1 9683  df-rank 9684

This theorem is referenced by: (None)