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Theorem pwfseq 10679

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 8961	. . 3 ⊢ Rel ≼
2	1	brrelex2i 5729	. 2 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
3		domeng 8974	. . 3 ⊢ (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴)))
4		bren 8965	. . . . . 6 ⊢ (ω ≈ 𝑡 ↔ ∃ℎ ℎ:ω–1-1-onto→𝑡)
5		harcl 9574	. . . . . . . . . 10 ⊢ (har‘𝒫 𝐴) ∈ On
6		infxpenc2 10037	. . . . . . . . . 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 7422	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (𝐴 ↑_m 𝑛) = (𝐴 ↑_m 𝑘))
10	9	cbviunv 5037	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) = ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘)
11		f1eq3 6784	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 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 4004	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19		fveq2 6891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑚‘𝑏) = (𝑚‘𝑤))
20	19	f1oeq1d 6828	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏))
21		xpeq12 5697	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 = 𝑤 ∧ 𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
22	21	anidms 566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
23	22	f1oeq2d 6829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝑤 → ((𝑚‘𝑤):(𝑏 × 𝑏)–1-1-onto→𝑏 ↔ (𝑚‘𝑤):(𝑤 × 𝑤)–1-1-onto→𝑏))
24		f1oeq3 6823	. . . . . . . . . . . . . . . . . . 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 3229	. . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . 15 ⊢ OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
30		eqid 2727	. . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . 15 ⊢ seq_ω((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)‘∅)⟩}) = seq_ω((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 7422	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑢 ↑_m 𝑛) = (𝑢 ↑_m 𝑘))
34	33	cbviunv 5037	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑛 ∈ ω (𝑢 ↑_m 𝑛) = ∪ 𝑘 ∈ ω (𝑢 ↑_m 𝑘)
35	34	mpteq1i 5238	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑢 ↑_m 𝑛) ↦ ⟨dom 𝑦, ((seq_ω((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, ((seq_ω((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 2727	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦 ∈ 𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
37		eqid 2727	. . . . . . . . . . . . . . 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, ((seq_ω((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, ((seq_ω((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 10678	. . . . . . . . . . . . . 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 1932	. . . . . . . . . . . 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 8970	. . . . . . . . . . . 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 1929	. . . . . . . . 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 1926	. . . . . 6 ⊢ (∃ℎ ℎ:ω–1-1-onto→𝑡 → (𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛)))
48	4, 47	sylbi 216	. . . . 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 1926	. . 3 ⊢ (∃𝑡(ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
51	3, 50	biimtrdi 252	. 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))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4598 {csn 4624 ⟨cop 4630 ∪ ciun 4991 class class class wbr 5142 ↦ cmpt 5225 We wwe 5626 × cxp 5670 ^◡ccnv 5671 dom cdm 5672 ↾ cres 5674 ∘ ccom 5676 Oncon0 6363 suc csuc 6365 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ωcom 7864 seq_ωcseqom 8461 ↑_m cmap 8836 ≈ cen 8952 ≼ cdom 8953 OrdIsocoi 9524 harchar 9571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-seqom 8462 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-oexp 8486 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-oi 9525 df-har 9572 df-cnf 9677 df-card 9954

This theorem is referenced by: pwxpndom2 10680

W3C validator