Theorem pwfseq 10547

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 (𝐴 ↑m 𝑛))

Distinct variable group:   𝐴,𝑛

Proof of Theorem pwfseq

Dummy variables 𝑓 𝑏 𝑔 ℎ 𝑘 𝑚 𝑝 𝑟 𝑠 𝑡 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldom 8870	. . 3 ⊢ Rel ≼
2	1	brrelex2i 5671	. 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		domeng 8880	. . 3 ⊢ (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴)))
4		bren 8874	. . . . . 6 ⊢ (ω ≈ 𝑡 ↔ ∃ℎ ℎ:ω–1-1-onto→𝑡)
5		harcl 9440	. . . . . . . . . 10 ⊢ (har‘𝒫 𝐴) ∈ On
6		infxpenc2 9905	. . . . . . . . . 10 ⊢ ((har‘𝒫 𝐴) ∈ On → ∃𝑚∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑚∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)
8		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
9		oveq2 7349	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑘))
10	9	cbviunv 4987	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)
11		f1eq3 6712	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) → (𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ 𝑔:𝒫 𝐴–1-1→∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ 𝑔:𝒫 𝐴–1-1→∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
13	8, 12	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝑔:𝒫 𝐴–1-1→∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
14		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → 𝑡 ⊆ 𝐴)
15		simplll 774	. . . . . . . . . . . . . . 15 ⊢ ((((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ℎ:ω–1-1-onto→𝑡)
16		biid 261	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑢 × 𝑢) ∧ 𝑟 We 𝑢) ∧ ω ≼ 𝑢) ↔ ((𝑢 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑢 × 𝑢) ∧ 𝑟 We 𝑢) ∧ ω ≼ 𝑢))
17		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
18		sseq2 3959	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19		fveq2 6817	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑚‘𝑏) = (𝑚‘𝑤))
20	19	f1oeq1d 6754	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏))
21		xpeq12 5639	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝑤 ∧ 𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
22	21	anidms 566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
23	22	f1oeq2d 6755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑏))
24		f1oeq3 6749	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑤))
25	20, 23, 24	3bitrd 305	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑤))
26	18, 25	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑤 → ((ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ (ω ⊆ 𝑤 → (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑤)))
27	26	cbvralvw 3208	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) ↔ ∀𝑤 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑤 → (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑤))
28	17, 27	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ∀𝑤 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑤 → (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑤))
29		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
30		eqid 2730	. . . . . . . . . . . . . . 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)⟩)
31		eqid 2730	. . . . . . . . . . . . . . 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)⟩))
32		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢 ↑m suc 𝑝) ↦ ((𝑓‘(𝑥 ↾ 𝑝))((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‘𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩}) = seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢 ↑m suc 𝑝) ↦ ((𝑓‘(𝑥 ↾ 𝑝))((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‘𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})
33		oveq2 7349	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑢 ↑m 𝑛) = (𝑢 ↑m 𝑘))
34	33	cbviunv 4987	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝑢 ↑m 𝑘)
35	34	mpteq1i 5180	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢 ↑m suc 𝑝) ↦ ((𝑓‘(𝑥 ↾ 𝑝))((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‘𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩) = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝑢 ↑m 𝑘) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢 ↑m suc 𝑝) ↦ ((𝑓‘(𝑥 ↾ 𝑝))((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‘𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩)
36		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
37		eqid 2730	. . . . . . . . . . . . . . 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, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)) ∘ (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢 ↑m suc 𝑝) ↦ ((𝑓‘(𝑥 ↾ 𝑝))((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‘𝑝)))), {⟨∅, (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, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)) ∘ (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢 ↑m suc 𝑝) ↦ ((𝑓‘(𝑥 ↾ 𝑝))((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‘𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩))
38	13, 14, 15, 16, 28, 29, 30, 31, 32, 35, 36, 37	pwfseqlem5 10546	. . . . . . . . . . . . . 14 ⊢ ¬ (((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) ∧ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
39	38	imnani 400	. . . . . . . . . . . . 13 ⊢ (((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) → ¬ 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
40	39	nexdv 1937	. . . . . . . . . . . 12 ⊢ (((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) → ¬ ∃𝑔 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
41		brdomi 8877	. . . . . . . . . . . 12 ⊢ (𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → ∃𝑔 𝑔:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
42	40, 41	nsyl 140	. . . . . . . . . . 11 ⊢ (((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
43	42	ex 412	. . . . . . . . . 10 ⊢ ((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) → (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
44	43	exlimdv 1934	. . . . . . . . 9 ⊢ ((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) → (∃𝑚∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
45	7, 44	mpi 20	. . . . . . . 8 ⊢ ((ℎ:ω–1-1-onto→𝑡 ∧ 𝑡 ⊆ 𝐴) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
46	45	ex 412	. . . . . . 7 ⊢ (ℎ:ω–1-1-onto→𝑡 → (𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
47	46	exlimiv 1931	. . . . . 6 ⊢ (∃ℎ ℎ:ω–1-1-onto→𝑡 → (𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
48	4, 47	sylbi 217	. . . . 5 ⊢ (ω ≈ 𝑡 → (𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
49	48	imp 406	. . . 4 ⊢ ((ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
50	49	exlimiv 1931	. . 3 ⊢ (∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
51	3, 50	biimtrdi 253	. 2 ⊢ (𝐴 ∈ V → (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)))
52	2, 51	mpcom 38	1 ⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4548  {csn 4574  ⟨cop 4580  ∪ ciun 4939   class class class wbr 5089   ↦ cmpt 5170   We wwe 5566   × cxp 5612  ^◡ccnv 5613  dom cdm 5614   ↾ cres 5616   ∘ ccom 5618  Oncon0 6302  suc csuc 6304  –1-1→wf1 6474  –1-1-onto→wf1o 6476  ‘cfv 6477  (class class class)co 7341   ∈ cmpo 7343  ωcom 7791  seq_ωcseqom 8361   ↑_m cmap 8745   ≈ cen 8861   ≼ cdom 8862  OrdIsocoi 9390  harchar 9437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  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-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-seqom 8362  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385  df-oexp 8386  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-oi 9391  df-har 9438  df-cnf 9547  df-card 9824

This theorem is referenced by:  pwxpndom2  10548