Theorem wfaxpow 45986

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 45948	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . . 5 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5219	. . . . 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 234	. . 3 ⊢ Tr 𝑊
6		pwclaxpow 45973	. . 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 703	. 2 ⊢ (∀𝑥 ∈ 𝑊 (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 → ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8		pwwf 9815	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 219	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
10		r1elssi 9813	. . . . 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 2849	. . . . . 6 ⊢ ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥 → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
13	11, 12	sylbi 220	. . . . 5 ⊢ (𝒫 𝑥 ⊆ ∪ (𝑅1 “ On) → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
14	9, 10, 13	3syl 19	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ((𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)))
15	9, 14	mpbird 260	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On))
16	2	eleq2i 2853	. . 3 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
17	2	ineq2i 4163	. . . 4 ⊢ (𝒫 𝑥 ∩ 𝑊) = (𝒫 𝑥 ∩ ∪ (𝑅1 “ On))
18	17, 2	eleq12i 2854	. . 3 ⊢ ((𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) ∈ ∪ (𝑅1 “ On))
19	15, 16, 18	3imtr4i 295	. 2 ⊢ (𝑥 ∈ 𝑊 → (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊)
20	7, 19	mprg 3083	1 ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  Tr wtr 5212   “ cima 5654  Oncon0 6362  𝑅₁cr1 9766

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769

This theorem is used by: (None)