Theorem pwfseq 10733

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 9009	. . 3 ⊢ Rel ≼
2	1	brrelex2i 5757	. 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		domeng 9022	. . 3 ⊢ (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴)))
4		bren 9013	. . . . . 6 ⊢ (ω ≈ 𝑡 ↔ ∃ℎ ℎ:ω–1-1-onto→𝑡)
5		harcl 9628	. . . . . . . . . 10 ⊢ (har‘𝒫 𝐴) ∈ On
6		infxpenc2 10091	. . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑘))
10	9	cbviunv 5063	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)
11		f1eq3 6814	. . . . . . . . . . . . . . . . 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 4035	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19		fveq2 6920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑚‘𝑏) = (𝑚‘𝑤))
20	19	f1oeq1d 6857	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏))
21		xpeq12 5725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝑤 ∧ 𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
22	21	anidms 566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
23	22	f1oeq2d 6858	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑏))
24		f1oeq3 6852	. . . . . . . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 15 ⊢ OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
30		eqid 2740	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑢 ↑m 𝑛) = (𝑢 ↑m 𝑘))
34	33	cbviunv 5063	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝑢 ↑m 𝑘)
35	34	mpteq1i 5262	. . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
37		eqid 2740	. . . . . . . . . . . . . . 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 10732	. . . . . . . . . . . . . 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 1935	. . . . . . . . . . . 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 9018	. . . . . . . . . . . 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 1932	. . . . . . . . 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 1929	. . . . . 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 1929	. . 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 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ⟨cop 4654  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249   We wwe 5651   × cxp 5698  ^◡ccnv 5699  dom cdm 5700   ↾ cres 5702   ∘ ccom 5704  Oncon0 6395  suc csuc 6397  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903  seq_ωcseqom 8503   ↑_m cmap 8884   ≈ cen 9000   ≼ cdom 9001  OrdIsocoi 9578  harchar 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqom 8504  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-oexp 8528  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-oi 9579  df-har 9626  df-cnf 9731  df-card 10008

This theorem is referenced by:  pwxpndom2  10734