Theorem pwfseq 10587

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 8901	. . 3 ⊢ Rel ≼
2	1	brrelex2i 5689	. 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		domeng 8911	. . 3 ⊢ (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴)))
4		bren 8905	. . . . . 6 ⊢ (ω ≈ 𝑡 ↔ ∃ℎ ℎ:ω–1-1-onto→𝑡)
5		harcl 9476	. . . . . . . . . 10 ⊢ (har‘𝒫 𝐴) ∈ On
6		infxpenc2 9944	. . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑘))
10	9	cbviunv 4996	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)
11		f1eq3 6735	. . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 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)) → 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 769	. . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19		fveq2 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑚‘𝑏) = (𝑚‘𝑤))
20	19	f1oeq1d 6777	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏))
21		xpeq12 5657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝑤 ∧ 𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
22	21	anidms 566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
23	22	f1oeq2d 6778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑏))
24		f1oeq3 6772	. . . . . . . . . . . . . . . . . . 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 3216	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
30		eqid 2737	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑢 ↑m 𝑛) = (𝑢 ↑m 𝑘))
34	33	cbviunv 4996	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝑢 ↑m 𝑘)
35	34	mpteq1i 5191	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
37		eqid 2737	. . . . . . . . . . . . . . 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 10586	. . . . . . . . . . . . . 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 1938	. . . . . . . . . . . 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 8908	. . . . . . . . . . . 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 1935	. . . . . . . . 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 1932	. . . . . 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 1932	. . 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 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ⟨cop 4588  ∪ ciun 4948   class class class wbr 5100   ↦ cmpt 5181   We wwe 5584   × cxp 5630  ^◡ccnv 5631  dom cdm 5632   ↾ cres 5634   ∘ ccom 5636  Oncon0 6325  suc csuc 6327  –1-1→wf1 6497  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  ωcom 7818  seq_ωcseqom 8388   ↑_m cmap 8775   ≈ cen 8892   ≼ cdom 8893  OrdIsocoi 9426  harchar 9473

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

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-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-seqom 8389  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412  df-oexp 8413  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-har 9474  df-cnf 9583  df-card 9863

This theorem is referenced by:  pwxpndom2  10588