Theorem wfaxpow 44980

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 44942	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . . 5 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5224	. . . . 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 44967	. . 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 9766	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 216	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
10		r1elssi 9764	. . . . 5 ⊢ (𝒫 𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ⊆ ∪ (𝑅1 “ On))
11		dfss2 3934	. . . . . 6 ⊢ (𝒫 𝑥 ⊆ ∪ (𝑅1 “ On) ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥)
12		eleq1 2817	. . . . . 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 2821	. . 3 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
17	2	ineq2i 4182	. . . 4 ⊢ (𝒫 𝑥 ∩ 𝑊) = (𝒫 𝑥 ∩ ∪ (𝑅1 “ On))
18	17, 2	eleq12i 2822	. . 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 3051	1 ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3915   ⊆ wss 3916  𝒫 cpw 4565  ∪ cuni 4873  Tr wtr 5216   “ cima 5643  Oncon0 6334  𝑅₁cr1 9721

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-om 7845  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-r1 9723  df-rank 9724

This theorem is referenced by: (None)