Theorem wfaxpow 45445

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 45407	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . . 5 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5200	. . . . 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 45432	. . 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 691	. 2 ⊢ (∀𝑥 ∈ 𝑊 (𝒫 𝑥 ∩ 𝑊) ∈ 𝑊 → ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8		pwwf 9725	. . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
9	8	biimpi 216	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On))
10		r1elssi 9723	. . . . 5 ⊢ (𝒫 𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ⊆ ∪ (𝑅1 “ On))
11		dfss2 3908	. . . . . 6 ⊢ (𝒫 𝑥 ⊆ ∪ (𝑅1 “ On) ↔ (𝒫 𝑥 ∩ ∪ (𝑅1 “ On)) = 𝒫 𝑥)
12		eleq1 2825	. . . . . 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 2829	. . 3 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
17	2	ineq2i 4158	. . . 4 ⊢ (𝒫 𝑥 ∩ 𝑊) = (𝒫 𝑥 ∩ ∪ (𝑅1 “ On))
18	17, 2	eleq12i 2830	. . 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 3058	1 ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  Tr wtr 5193   “ cima 5628  Oncon0 6318  𝑅₁cr1 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9682  df-rank 9683

This theorem is referenced by: (None)