pw1nct - Mathbox for Jim Kingdon

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Theorem pw1nct 14722

Description: A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)

**Assertion**
Ref	Expression
pw1nct	⊢ (∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1_o ⊔ 1_o))

Distinct variable groups: 𝑓,𝑚,𝑛,𝑝,𝑟 𝑓,𝑞,𝑚,𝑟

Proof of Theorem pw1nct Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1528	. . . . . . . 8 ⊢ Ⅎ𝑚 𝑟 ⊆ (𝒫 1_o × ω)
2		nfv 1528	. . . . . . . . 9 ⊢ Ⅎ𝑚∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛
3		nfre1 2520	. . . . . . . . 9 ⊢ Ⅎ𝑚∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚
4	2, 3	nfim 1572	. . . . . . . 8 ⊢ Ⅎ𝑚(∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)
5	1, 4	nfim 1572	. . . . . . 7 ⊢ Ⅎ𝑚(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚))
6	5	nfal 1576	. . . . . 6 ⊢ Ⅎ𝑚∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚))
7		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑚 𝑓:ω–onto→𝒫 1_o
8	6, 7	nfan 1565	. . . . 5 ⊢ Ⅎ𝑚(∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o)
9		breq1 4006	. . . . . . . . 9 ⊢ (𝑞 = ∅ → (𝑞^◡𝑓𝑚 ↔ ∅^◡𝑓𝑚))
10		simpr 110	. . . . . . . . 9 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚)
11		0elpw 4164	. . . . . . . . . 10 ⊢ ∅ ∈ 𝒫 1_o
12	11	a1i 9	. . . . . . . . 9 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ∅ ∈ 𝒫 1_o)
13	9, 10, 12	rspcdva 2846	. . . . . . . 8 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ∅^◡𝑓𝑚)
14		0ex 4130	. . . . . . . . 9 ⊢ ∅ ∈ V
15		vex 2740	. . . . . . . . 9 ⊢ 𝑚 ∈ V
16	14, 15	brcnv 4810	. . . . . . . 8 ⊢ (∅^◡𝑓𝑚 ↔ 𝑚𝑓∅)
17	13, 16	sylib 122	. . . . . . 7 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → 𝑚𝑓∅)
18		fofn 5440	. . . . . . . . 9 ⊢ (𝑓:ω–onto→𝒫 1_o → 𝑓 Fn ω)
19	18	ad3antlr 493	. . . . . . . 8 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → 𝑓 Fn ω)
20		simplr 528	. . . . . . . 8 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → 𝑚 ∈ ω)
21		fnbrfvb 5556	. . . . . . . 8 ⊢ ((𝑓 Fn ω ∧ 𝑚 ∈ ω) → ((𝑓‘𝑚) = ∅ ↔ 𝑚𝑓∅))
22	19, 20, 21	syl2anc 411	. . . . . . 7 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ((𝑓‘𝑚) = ∅ ↔ 𝑚𝑓∅))
23	17, 22	mpbird 167	. . . . . 6 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → (𝑓‘𝑚) = ∅)
24		breq1 4006	. . . . . . . . . 10 ⊢ (𝑞 = 1_o → (𝑞^◡𝑓𝑚 ↔ 1_o^◡𝑓𝑚))
25		1oex 6424	. . . . . . . . . . . 12 ⊢ 1_o ∈ V
26	25	pwid 3590	. . . . . . . . . . 11 ⊢ 1_o ∈ 𝒫 1_o
27	26	a1i 9	. . . . . . . . . 10 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → 1_o ∈ 𝒫 1_o)
28	24, 10, 27	rspcdva 2846	. . . . . . . . 9 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → 1_o^◡𝑓𝑚)
29	25, 15	brcnv 4810	. . . . . . . . 9 ⊢ (1_o^◡𝑓𝑚 ↔ 𝑚𝑓1_o)
30	28, 29	sylib 122	. . . . . . . 8 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → 𝑚𝑓1_o)
31		fnbrfvb 5556	. . . . . . . . 9 ⊢ ((𝑓 Fn ω ∧ 𝑚 ∈ ω) → ((𝑓‘𝑚) = 1_o ↔ 𝑚𝑓1_o))
32	19, 20, 31	syl2anc 411	. . . . . . . 8 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ((𝑓‘𝑚) = 1_o ↔ 𝑚𝑓1_o))
33	30, 32	mpbird 167	. . . . . . 7 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → (𝑓‘𝑚) = 1_o)
34		1n0 6432	. . . . . . . 8 ⊢ 1_o ≠ ∅
35	34	neii 2349	. . . . . . 7 ⊢ ¬ 1_o = ∅
36		eqeq1 2184	. . . . . . . . 9 ⊢ ((𝑓‘𝑚) = 1_o → ((𝑓‘𝑚) = ∅ ↔ 1_o = ∅))
37	36	biimpd 144	. . . . . . . 8 ⊢ ((𝑓‘𝑚) = 1_o → ((𝑓‘𝑚) = ∅ → 1_o = ∅))
38	37	con3dimp 635	. . . . . . 7 ⊢ (((𝑓‘𝑚) = 1_o ∧ ¬ 1_o = ∅) → ¬ (𝑓‘𝑚) = ∅)
39	33, 35, 38	sylancl 413	. . . . . 6 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ¬ (𝑓‘𝑚) = ∅)
40	23, 39	pm2.21fal 1373	. . . . 5 ⊢ ((((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) ∧ 𝑚 ∈ ω) ∧ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚) → ⊥)
41		fof 5438	. . . . . . . 8 ⊢ (𝑓:ω–onto→𝒫 1_o → 𝑓:ω⟶𝒫 1_o)
42		fssxp 5383	. . . . . . . . . 10 ⊢ (𝑓:ω⟶𝒫 1_o → 𝑓 ⊆ (ω × 𝒫 1_o))
43		cnvss 4800	. . . . . . . . . 10 ⊢ (𝑓 ⊆ (ω × 𝒫 1_o) → ^◡𝑓 ⊆ ^◡(ω × 𝒫 1_o))
44	42, 43	syl 14	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 1_o → ^◡𝑓 ⊆ ^◡(ω × 𝒫 1_o))
45		cnvxp 5047	. . . . . . . . 9 ⊢ ^◡(ω × 𝒫 1_o) = (𝒫 1_o × ω)
46	44, 45	sseqtrdi 3203	. . . . . . . 8 ⊢ (𝑓:ω⟶𝒫 1_o → ^◡𝑓 ⊆ (𝒫 1_o × ω))
47	41, 46	syl 14	. . . . . . 7 ⊢ (𝑓:ω–onto→𝒫 1_o → ^◡𝑓 ⊆ (𝒫 1_o × ω))
48	47	adantl 277	. . . . . 6 ⊢ ((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) → ^◡𝑓 ⊆ (𝒫 1_o × ω))
49		foelrn 5753	. . . . . . . . 9 ⊢ ((𝑓:ω–onto→𝒫 1_o ∧ 𝑝 ∈ 𝒫 1_o) → ∃𝑛 ∈ ω 𝑝 = (𝑓‘𝑛))
50	18	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝑓:ω–onto→𝒫 1_o ∧ 𝑝 ∈ 𝒫 1_o) ∧ 𝑛 ∈ ω) → 𝑓 Fn ω)
51		simpr 110	. . . . . . . . . . 11 ⊢ (((𝑓:ω–onto→𝒫 1_o ∧ 𝑝 ∈ 𝒫 1_o) ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
52		eqcom 2179	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑛) = 𝑝 ↔ 𝑝 = (𝑓‘𝑛))
53		fnbrfvb 5556	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn ω ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) = 𝑝 ↔ 𝑛𝑓𝑝))
54		brcnvg 4808	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ V ∧ 𝑛 ∈ V) → (𝑝^◡𝑓𝑛 ↔ 𝑛𝑓𝑝))
55	54	elvd 2742	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ V → (𝑝^◡𝑓𝑛 ↔ 𝑛𝑓𝑝))
56	55	elv 2741	. . . . . . . . . . . . 13 ⊢ (𝑝^◡𝑓𝑛 ↔ 𝑛𝑓𝑝)
57	53, 56	bitr4di 198	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn ω ∧ 𝑛 ∈ ω) → ((𝑓‘𝑛) = 𝑝 ↔ 𝑝^◡𝑓𝑛))
58	52, 57	bitr3id 194	. . . . . . . . . . 11 ⊢ ((𝑓 Fn ω ∧ 𝑛 ∈ ω) → (𝑝 = (𝑓‘𝑛) ↔ 𝑝^◡𝑓𝑛))
59	50, 51, 58	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑓:ω–onto→𝒫 1_o ∧ 𝑝 ∈ 𝒫 1_o) ∧ 𝑛 ∈ ω) → (𝑝 = (𝑓‘𝑛) ↔ 𝑝^◡𝑓𝑛))
60	59	rexbidva 2474	. . . . . . . . 9 ⊢ ((𝑓:ω–onto→𝒫 1_o ∧ 𝑝 ∈ 𝒫 1_o) → (∃𝑛 ∈ ω 𝑝 = (𝑓‘𝑛) ↔ ∃𝑛 ∈ ω 𝑝^◡𝑓𝑛))
61	49, 60	mpbid 147	. . . . . . . 8 ⊢ ((𝑓:ω–onto→𝒫 1_o ∧ 𝑝 ∈ 𝒫 1_o) → ∃𝑛 ∈ ω 𝑝^◡𝑓𝑛)
62	61	ralrimiva 2550	. . . . . . 7 ⊢ (𝑓:ω–onto→𝒫 1_o → ∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛)
63	62	adantl 277	. . . . . 6 ⊢ ((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) → ∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛)
64		cnvexg 5166	. . . . . . . 8 ⊢ (𝑓 ∈ V → ^◡𝑓 ∈ V)
65	64	elv 2741	. . . . . . 7 ⊢ ^◡𝑓 ∈ V
66		simpl 109	. . . . . . 7 ⊢ ((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) → ∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)))
67		sseq1 3178	. . . . . . . . 9 ⊢ (𝑟 = ^◡𝑓 → (𝑟 ⊆ (𝒫 1_o × ω) ↔ ^◡𝑓 ⊆ (𝒫 1_o × ω)))
68		breq 4005	. . . . . . . . . . . 12 ⊢ (𝑟 = ^◡𝑓 → (𝑝𝑟𝑛 ↔ 𝑝^◡𝑓𝑛))
69	68	rexbidv 2478	. . . . . . . . . . 11 ⊢ (𝑟 = ^◡𝑓 → (∃𝑛 ∈ ω 𝑝𝑟𝑛 ↔ ∃𝑛 ∈ ω 𝑝^◡𝑓𝑛))
70	69	ralbidv 2477	. . . . . . . . . 10 ⊢ (𝑟 = ^◡𝑓 → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 ↔ ∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛))
71		breq 4005	. . . . . . . . . . . 12 ⊢ (𝑟 = ^◡𝑓 → (𝑞𝑟𝑚 ↔ 𝑞^◡𝑓𝑚))
72	71	ralbidv 2477	. . . . . . . . . . 11 ⊢ (𝑟 = ^◡𝑓 → (∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚 ↔ ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚))
73	72	rexbidv 2478	. . . . . . . . . 10 ⊢ (𝑟 = ^◡𝑓 → (∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚 ↔ ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚))
74	70, 73	imbi12d 234	. . . . . . . . 9 ⊢ (𝑟 = ^◡𝑓 → ((∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚) ↔ (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚)))
75	67, 74	imbi12d 234	. . . . . . . 8 ⊢ (𝑟 = ^◡𝑓 → ((𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ↔ (^◡𝑓 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚))))
76	75	spcgv 2824	. . . . . . 7 ⊢ (^◡𝑓 ∈ V → (∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) → (^◡𝑓 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚))))
77	65, 66, 76	mpsyl 65	. . . . . 6 ⊢ ((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) → (^◡𝑓 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝^◡𝑓𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚)))
78	48, 63, 77	mp2d 47	. . . . 5 ⊢ ((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞^◡𝑓𝑚)
79	8, 40, 78	r19.29af 2618	. . . 4 ⊢ ((∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) ∧ 𝑓:ω–onto→𝒫 1_o) → ⊥)
80	79	inegd 1372	. . 3 ⊢ (∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) → ¬ 𝑓:ω–onto→𝒫 1_o)
81	80	nexdv 1936	. 2 ⊢ (∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→𝒫 1_o)
82		elex2 2753	. . 3 ⊢ (∅ ∈ 𝒫 1_o → ∃𝑤 𝑤 ∈ 𝒫 1_o)
83		ctm 7107	. . 3 ⊢ (∃𝑤 𝑤 ∈ 𝒫 1_o → (∃𝑓 𝑓:ω–onto→(𝒫 1_o ⊔ 1_o) ↔ ∃𝑓 𝑓:ω–onto→𝒫 1_o))
84	11, 82, 83	mp2b 8	. 2 ⊢ (∃𝑓 𝑓:ω–onto→(𝒫 1_o ⊔ 1_o) ↔ ∃𝑓 𝑓:ω–onto→𝒫 1_o)
85	81, 84	sylnibr 677	1 ⊢ (∀𝑟(𝑟 ⊆ (𝒫 1_o × ω) → (∀𝑝 ∈ 𝒫 1_o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1_o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1_o ⊔ 1_o))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 = wceq 1353 ⊥wfal 1358 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 Vcvv 2737 ⊆ wss 3129 ∅c0 3422 𝒫 cpw 3575 class class class wbr 4003 ωcom 4589 × cxp 4624 ^◡ccnv 4625 Fn wfn 5211 ⟶wf 5212 –onto→wfo 5214 ‘cfv 5216 1_oc1o 6409 ⊔ cdju 7035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587

This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-1st 6140 df-2nd 6141 df-1o 6416 df-dju 7036 df-inl 7045 df-inr 7046 df-case 7082

This theorem is referenced by: (None)

W3C validator