Theorem pwfseq 10683

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 8970	. . 3 ⊢ Rel ≼
2	1	brrelex2i 5716	. 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		domeng 8982	. . 3 ⊢ (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴)))
4		bren 8974	. . . . . 6 ⊢ (ω ≈ 𝑡 ↔ ∃ℎ ℎ:ω–1-1-onto→𝑡)
5		harcl 9578	. . . . . . . . . 10 ⊢ (har‘𝒫 𝐴) ∈ On
6		infxpenc2 10041	. . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑘))
10	9	cbviunv 5021	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)
11		f1eq3 6776	. . . . . . . . . . . . . . . . 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 3990	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19		fveq2 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑚‘𝑏) = (𝑚‘𝑤))
20	19	f1oeq1d 6818	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏))
21		xpeq12 5684	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝑤 ∧ 𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
22	21	anidms 566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
23	22	f1oeq2d 6819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑏))
24		f1oeq3 6813	. . . . . . . . . . . . . . . . . . 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 3224	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
30		eqid 2736	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑢 ↑m 𝑛) = (𝑢 ↑m 𝑘))
34	33	cbviunv 5021	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ ω (𝑢 ↑m 𝑛) = ∪ 𝑘 ∈ ω (𝑢 ↑m 𝑘)
35	34	mpteq1i 5216	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
37		eqid 2736	. . . . . . . . . . . . . . 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 10682	. . . . . . . . . . . . . 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 1936	. . . . . . . . . . . 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 8978	. . . . . . . . . . . 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 1933	. . . . . . . . 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 1930	. . . . . 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 1930	. . 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 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ⟨cop 4612  ∪ ciun 4972   class class class wbr 5124   ↦ cmpt 5206   We wwe 5610   × cxp 5657  ^◡ccnv 5658  dom cdm 5659   ↾ cres 5661   ∘ ccom 5663  Oncon0 6357  suc csuc 6359  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  ωcom 7866  seq_ωcseqom 8466   ↑_m cmap 8845   ≈ cen 8961   ≼ cdom 8962  OrdIsocoi 9528  harchar 9575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seqom 8467  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490  df-oexp 8491  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-oi 9529  df-har 9576  df-cnf 9681  df-card 9958

This theorem is referenced by:  pwxpndom2  10684