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Theorem pwfseq 9365
 Description: The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
pwfseq (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
Distinct variable group:   𝐴,𝑛

Proof of Theorem pwfseq
Dummy variables 𝑓 𝑏 𝑔 𝑘 𝑚 𝑝 𝑟 𝑠 𝑡 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 7847 . . 3 Rel ≼
21brrelex2i 5083 . 2 (ω ≼ 𝐴𝐴 ∈ V)
3 domeng 7855 . . 3 (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡𝑡𝐴)))
4 bren 7850 . . . . . 6 (ω ≈ 𝑡 ↔ ∃ :ω–1-1-onto𝑡)
5 harcl 8349 . . . . . . . . . 10 (har‘𝒫 𝐴) ∈ On
6 infxpenc2 8728 . . . . . . . . . 10 ((har‘𝒫 𝐴) ∈ On → ∃𝑚𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
75, 6ax-mp 5 . . . . . . . . 9 𝑚𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)
8 simpr 476 . . . . . . . . . . . . . . . 16 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
9 oveq2 6557 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → (𝐴𝑚 𝑛) = (𝐴𝑚 𝑘))
109cbviunv 4495 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ω (𝐴𝑚 𝑛) = 𝑘 ∈ ω (𝐴𝑚 𝑘)
11 f1eq3 6011 . . . . . . . . . . . . . . . . 17 ( 𝑛 ∈ ω (𝐴𝑚 𝑛) = 𝑘 ∈ ω (𝐴𝑚 𝑘) → (𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ 𝑔:𝒫 𝐴1-1 𝑘 ∈ ω (𝐴𝑚 𝑘)))
1210, 11ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ 𝑔:𝒫 𝐴1-1 𝑘 ∈ ω (𝐴𝑚 𝑘))
138, 12sylib 207 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝑔:𝒫 𝐴1-1 𝑘 ∈ ω (𝐴𝑚 𝑘))
14 simpllr 795 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛)) → 𝑡𝐴)
15 simplll 794 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛)) → :ω–1-1-onto𝑡)
16 biid 250 . . . . . . . . . . . . . . 15 (((𝑢𝐴𝑟 ⊆ (𝑢 × 𝑢) ∧ 𝑟 We 𝑢) ∧ ω ≼ 𝑢) ↔ ((𝑢𝐴𝑟 ⊆ (𝑢 × 𝑢) ∧ 𝑟 We 𝑢) ∧ ω ≼ 𝑢))
17 simplr 788 . . . . . . . . . . . . . . . 16 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛)) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
18 sseq2 3590 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑤 → (𝑚𝑏) = (𝑚𝑤))
20 f1oeq1 6040 . . . . . . . . . . . . . . . . . . . 20 ((𝑚𝑏) = (𝑚𝑤) → ((𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑏 × 𝑏)–1-1-onto𝑏))
2119, 20syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑤 → ((𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑏 × 𝑏)–1-1-onto𝑏))
22 xpeq12 5058 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑤𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
2322anidms 675 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
24 f1oeq2 6041 . . . . . . . . . . . . . . . . . . . 20 ((𝑏 × 𝑏) = (𝑤 × 𝑤) → ((𝑚𝑤):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑏))
2523, 24syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑤 → ((𝑚𝑤):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑏))
26 f1oeq3 6042 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑤 → ((𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
2721, 25, 263bitrd 293 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑤 → ((𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
2818, 27imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑤 → ((ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ (ω ⊆ 𝑤 → (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤)))
2928cbvralv 3147 . . . . . . . . . . . . . . . 16 (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ ∀𝑤 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑤 → (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
3017, 29sylib 207 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛)) → ∀𝑤 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑤 → (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
31 eqid 2610 . . . . . . . . . . . . . . 15 OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
32 eqid 2610 . . . . . . . . . . . . . . 15 (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩) = (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)
33 eqid 2610 . . . . . . . . . . . . . . 15 ((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)) = ((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))
34 eqid 2610 . . . . . . . . . . . . . . 15 seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩}) = seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})
35 oveq2 6557 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑢𝑚 𝑛) = (𝑢𝑚 𝑘))
3635cbviunv 4495 . . . . . . . . . . . . . . . 16 𝑛 ∈ ω (𝑢𝑚 𝑛) = 𝑘 ∈ ω (𝑢𝑚 𝑘)
37 mpteq1 4665 . . . . . . . . . . . . . . . 16 ( 𝑛 ∈ ω (𝑢𝑚 𝑛) = 𝑘 ∈ ω (𝑢𝑚 𝑘) → (𝑦 𝑛 ∈ ω (𝑢𝑚 𝑛) ↦ ⟨dom 𝑦, ((seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩) = (𝑦 𝑘 ∈ ω (𝑢𝑚 𝑘) ↦ ⟨dom 𝑦, ((seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩))
3836, 37ax-mp 5 . . . . . . . . . . . . . . 15 (𝑦 𝑛 ∈ ω (𝑢𝑚 𝑛) ↦ ⟨dom 𝑦, ((seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩) = (𝑦 𝑘 ∈ ω (𝑢𝑚 𝑘) ↦ ⟨dom 𝑦, ((seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩)
39 eqid 2610 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
40 eqid 2610 . . . . . . . . . . . . . . 15 ((((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)) ∘ (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)) ∘ (𝑦 𝑛 ∈ ω (𝑢𝑚 𝑛) ↦ ⟨dom 𝑦, ((seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩)) = ((((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)) ∘ (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)) ∘ (𝑦 𝑛 ∈ ω (𝑢𝑚 𝑛) ↦ ⟨dom 𝑦, ((seq𝜔((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢𝑚 suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩))
4113, 14, 15, 16, 30, 31, 32, 33, 34, 38, 39, 40pwfseqlem5 9364 . . . . . . . . . . . . . 14 ¬ (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
4241imnani 438 . . . . . . . . . . . . 13 (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) → ¬ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
4342nexdv 1851 . . . . . . . . . . . 12 (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) → ¬ ∃𝑔 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
44 brdomi 7852 . . . . . . . . . . . 12 (𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛) → ∃𝑔 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
4543, 44nsyl 134 . . . . . . . . . . 11 (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
4645ex 449 . . . . . . . . . 10 ((:ω–1-1-onto𝑡𝑡𝐴) → (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
4746exlimdv 1848 . . . . . . . . 9 ((:ω–1-1-onto𝑡𝑡𝐴) → (∃𝑚𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
487, 47mpi 20 . . . . . . . 8 ((:ω–1-1-onto𝑡𝑡𝐴) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
4948ex 449 . . . . . . 7 (:ω–1-1-onto𝑡 → (𝑡𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
5049exlimiv 1845 . . . . . 6 (∃ :ω–1-1-onto𝑡 → (𝑡𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
514, 50sylbi 206 . . . . 5 (ω ≈ 𝑡 → (𝑡𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
5251imp 444 . . . 4 ((ω ≈ 𝑡𝑡𝐴) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
5352exlimiv 1845 . . 3 (∃𝑡(ω ≈ 𝑡𝑡𝐴) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
543, 53syl6bi 242 . 2 (𝐴 ∈ V → (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛)))
552, 54mpcom 37 1 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   ∘ ccom 5042  Oncon0 5640  suc csuc 5642  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  seq𝜔cseqom 7429   ↑𝑚 cmap 7744   ≈ cen 7838   ≼ cdom 7839  OrdIsocoi 8297  harchar 8344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-har 8346  df-cnf 8442  df-card 8648 This theorem is referenced by:  pwxpndom2  9366
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