Theorem pwfseqlem4 10078

Description: Lemma for pwfseq 10080. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10076. Applying fpwwe2 10059 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 contradicts our assumption on 𝐹, so we are reduced to the case of 𝑍 finite. This too is a contradiction, though, because 𝑍 and its preimage under (𝑊‘𝑍) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses

Ref	Expression
pwfseqlem4.g	⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
pwfseqlem4.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem4.h	⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem4.ps	⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k	⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
pwfseqlem4.d	⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
pwfseqlem4.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
pwfseqlem4.w	⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}
pwfseqlem4.z	⊢ 𝑍 = ∪ dom 𝑊

Assertion

Ref	Expression
pwfseqlem4	⊢ ¬ 𝜑

Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑎,𝑏,𝑠,𝑣,𝐹   𝑤,𝐺   𝑤,𝐾   𝑟,𝑎,𝑥,𝑧,𝐻,𝑏,𝑠,𝑣   𝑛,𝑎,𝜑,𝑏,𝑠,𝑣,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑎,𝑛,𝑟,𝑠,𝑥,𝑧   𝑊,𝑎,𝑏,𝑠,𝑣   𝑍,𝑎,𝑏,𝑠,𝑣

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑣,𝑠,𝑟,𝑎,𝑏)   𝐴(𝑤,𝑣,𝑏)   𝐷(𝑥,𝑤,𝑣,𝑠,𝑟,𝑎,𝑏)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑧,𝑤,𝑛,𝑟)   𝑋(𝑥,𝑧,𝑤,𝑣,𝑛,𝑠,𝑟,𝑎,𝑏)   𝑍(𝑥,𝑧,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem4

Step	Hyp	Ref	Expression
1		eqid 2821	. . . . . . . . . . 11 ⊢ 𝑍 = 𝑍
2		eqid 2821	. . . . . . . . . . 11 ⊢ (𝑊‘𝑍) = (𝑊‘𝑍)
3	1, 2	pm3.2i 473	. . . . . . . . . 10 ⊢ (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍))
4		pwfseqlem4.w	. . . . . . . . . . 11 ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}
5		pwfseqlem4.g	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
6		omex 9100	. . . . . . . . . . . . . 14 ⊢ ω ∈ V
7		ovex 7183	. . . . . . . . . . . . . 14 ⊢ (𝐴 ↑m 𝑛) ∈ V
8	6, 7	iunex 7663	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V
9		f1dmex 7652	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ V) → 𝒫 𝐴 ∈ V)
10	5, 8, 9	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
11		pwexb 7482	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
12	10, 11	sylibr 236	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ V)
13		pwfseqlem4.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
14		pwfseqlem4.h	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
15		pwfseqlem4.ps	. . . . . . . . . . . 12 ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
16		pwfseqlem4.k	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
17		pwfseqlem4.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
18		pwfseqlem4.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
19	5, 13, 14, 15, 16, 17, 18	pwfseqlem4a 10077	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
20		pwfseqlem4.z	. . . . . . . . . . 11 ⊢ 𝑍 = ∪ dom 𝑊
21	4, 12, 19, 20	fpwwe2 10059	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍))))
22	3, 21	mpbiri 260	. . . . . . . . 9 ⊢ (𝜑 → (𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍))
23	22	simprd 498	. . . . . . . 8 ⊢ (𝜑 → (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍)
24	22	simpld 497	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑍𝑊(𝑊‘𝑍))
25	4, 12	fpwwe2lem2 10048	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍𝑊(𝑊‘𝑍) ↔ ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))))
26	24, 25	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)))
27	26	simpld 497	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)))
28	27	simpld 497	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
29	12, 28	ssexd 5220	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ V)
30		sseq1 3991	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑍 → (𝑎 ⊆ 𝐴 ↔ 𝑍 ⊆ 𝐴))
31		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑍 → 𝑎 = 𝑍)
32	31	sqxpeqd 5581	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍))
33	32	sseq2d 3998	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑍 → ((𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)))
34		weeq2 5538	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑍 → ((𝑊‘𝑍) We 𝑎 ↔ (𝑊‘𝑍) We 𝑍))
35	30, 33, 34	3anbi123d 1432	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑍 → ((𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎) ↔ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)))
36	35	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ (𝜑 ∧ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))))
37		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
38	37	3expa 1114	. . . . . . . . . . . . . . 15 ⊢ (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
39	38	adantrr 715	. . . . . . . . . . . . . 14 ⊢ (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
40	26, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))
41	40	pm4.71i 562	. . . . . . . . . . . 12 ⊢ (𝜑 ↔ (𝜑 ∧ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)))
42	36, 41	syl6bbr 291	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ 𝜑))
43		oveq1 7157	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑍 → (𝑎𝐹(𝑊‘𝑍)) = (𝑍𝐹(𝑊‘𝑍)))
44	43, 31	eleq12d 2907	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑍 → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍))
45		breq1 5061	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω))
46	44, 45	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑍 → (((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)))
47	42, 46	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω))))
48		fvex 6677	. . . . . . . . . . 11 ⊢ (𝑊‘𝑍) ∈ V
49		sseq1 3991	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑊‘𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎)))
50		weeq1 5537	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑊‘𝑍) → (𝑠 We 𝑎 ↔ (𝑊‘𝑍) We 𝑎))
51	49, 50	3anbi23d 1435	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑊‘𝑍) → ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)))
52	51	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑊‘𝑍) → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎))))
53		oveq2 7158	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑊‘𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊‘𝑍)))
54	53	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑊‘𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎))
55	54	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑊‘𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)))
56	52, 55	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑊‘𝑍) → (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω)) ↔ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω))))
57		omelon 9103	. . . . . . . . . . . . . . 15 ⊢ ω ∈ On
58		onenon 9372	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ On → ω ∈ dom card)
59	57, 58	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ω ∈ dom card
60		simpr3 1192	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎)
61	60	19.8ad 2177	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎)
62		ween 9455	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ dom card ↔ ∃𝑠 𝑠 We 𝑎)
63	61, 62	sylibr 236	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card)
64		domtri2 9412	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ dom card ∧ 𝑎 ∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
65	59, 63, 64	sylancr 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω))
66		nfv 1911	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))
67		nfcv 2977	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟𝑎
68		nfmpo2 7229	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
69	18, 68	nfcxfr 2975	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟𝐹
70		nfcv 2977	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟𝑠
71	67, 69, 70	nfov 7180	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑟(𝑎𝐹𝑠)
72	71	nfel1 2994	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)
73	66, 72	nfim 1893	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
74		sseq1 3991	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
75		weeq1 5537	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
76	74, 75	3anbi23d 1435	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)))
77	76	anbi1d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑠 → (((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))
78	77	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))))
79		oveq2 7158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠))
80	79	eleq1d 2897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)))
81	78, 80	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑠 → (((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))))
82		nfv 1911	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))
83		nfcv 2977	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝑎
84		nfmpo1 7228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
85	18, 84	nfcxfr 2975	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐹
86		nfcv 2977	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝑟
87	83, 85, 86	nfov 7180	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝑎𝐹𝑟)
88	87	nfel1 2994	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)
89	82, 88	nfim 1893	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
90		sseq1 3991	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
91		xpeq12 5574	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
92	91	anidms 569	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎))
93	92	sseq2d 3998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎)))
94		weeq2 5538	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
95	90, 93, 94	3anbi123d 1432	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎)))
96		breq2 5062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎))
97	95, 96	anbi12d 632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
98	15, 97	syl5bb 285	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))
99	98	anbi2d 630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))))
100		oveq1 7157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟))
101		difeq2 4092	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎))
102	100, 101	eleq12d 2907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)))
103	99, 102	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))))
104	5, 13, 14, 15, 16, 17, 18	pwfseqlem3 10076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
105	89, 103, 104	chvarfv 2238	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))
106	73, 81, 105	chvarfv 2238	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))
107	106	eldifbd 3948	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎)
108	107	expr 459	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
109	65, 108	sylbird 262	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → ¬ (𝑎𝐹𝑠) ∈ 𝑎))
110	109	con4d 115	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω))
111	48, 56, 110	vtocl 3559	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω))
112	47, 111	vtoclg 3567	. . . . . . . . 9 ⊢ (𝑍 ∈ V → (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)))
113	29, 112	mpcom 38	. . . . . . . 8 ⊢ (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω))
114	23, 113	mpd 15	. . . . . . 7 ⊢ (𝜑 → 𝑍 ≺ ω)
115		isfinite 9109	. . . . . . 7 ⊢ (𝑍 ∈ Fin ↔ 𝑍 ≺ ω)
116	114, 115	sylibr 236	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ Fin)
117	5, 13, 14, 15, 16, 17, 18	pwfseqlem2 10075	. . . . . 6 ⊢ ((𝑍 ∈ Fin ∧ (𝑊‘𝑍) ∈ V) → (𝑍𝐹(𝑊‘𝑍)) = (𝐻‘(card‘𝑍)))
118	116, 48, 117	sylancl 588	. . . . 5 ⊢ (𝜑 → (𝑍𝐹(𝑊‘𝑍)) = (𝐻‘(card‘𝑍)))
119	118, 23	eqeltrrd 2914	. . . 4 ⊢ (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍)
120	4, 12, 24	fpwwe2lem3 10049	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
121	119, 120	mpdan 685	. . . . . . . . 9 ⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍)))
122		cnvimass 5943	. . . . . . . . . . . 12 ⊢ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊‘𝑍)
123	27	simprd 498	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊‘𝑍) ⊆ (𝑍 × 𝑍))
124		dmss 5765	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍))
125	123, 124	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍))
126		dmxpss 6022	. . . . . . . . . . . . 13 ⊢ dom (𝑍 × 𝑍) ⊆ 𝑍
127	125, 126	sstrdi 3978	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝑊‘𝑍) ⊆ 𝑍)
128	122, 127	sstrid 3977	. . . . . . . . . . 11 ⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍)
129	116, 128	ssfid 8735	. . . . . . . . . 10 ⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin)
130	48	inex1 5213	. . . . . . . . . 10 ⊢ ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V
131	5, 13, 14, 15, 16, 17, 18	pwfseqlem2 10075	. . . . . . . . . 10 ⊢ (((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin ∧ ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V) → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
132	129, 130, 131	sylancl 588	. . . . . . . . 9 ⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
133	121, 132	eqtr3d 2858	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
134		f1of1 6608	. . . . . . . . . 10 ⊢ (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω–1-1→𝑋)
135	14, 134	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:ω–1-1→𝑋)
136		ficardom 9384	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (card‘𝑍) ∈ ω)
137	116, 136	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (card‘𝑍) ∈ ω)
138		ficardom 9384	. . . . . . . . . 10 ⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
139	129, 138	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)
140		f1fveq 7014	. . . . . . . . 9 ⊢ ((𝐻:ω–1-1→𝑋 ∧ ((card‘𝑍) ∈ ω ∧ (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)) → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
141	135, 137, 139, 140	syl12anc 834	. . . . . . . 8 ⊢ (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))))
142	133, 141	mpbid 234	. . . . . . 7 ⊢ (𝜑 → (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))
143	142	eqcomd 2827	. . . . . 6 ⊢ (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍))
144		finnum 9371	. . . . . . . 8 ⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
145	129, 144	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card)
146		finnum 9371	. . . . . . . 8 ⊢ (𝑍 ∈ Fin → 𝑍 ∈ dom card)
147	116, 146	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ dom card)
148		carden2 9410	. . . . . . 7 ⊢ (((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card ∧ 𝑍 ∈ dom card) → ((card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
149	145, 147, 148	syl2anc 586	. . . . . 6 ⊢ (𝜑 → ((card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
150	143, 149	mpbid 234	. . . . 5 ⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
151		dfpss2 4061	. . . . . . . 8 ⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
152	151	baib 538	. . . . . . 7 ⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
153	128, 152	syl 17	. . . . . 6 ⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍))
154		php3 8697	. . . . . . . . 9 ⊢ ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍)
155		sdomnen 8532	. . . . . . . . 9 ⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
156	154, 155	syl 17	. . . . . . . 8 ⊢ ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)
157	156	ex 415	. . . . . . 7 ⊢ (𝑍 ∈ Fin → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
158	116, 157	syl 17	. . . . . 6 ⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
159	153, 158	sylbird 262	. . . . 5 ⊢ (𝜑 → (¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍))
160	150, 159	mt4d 117	. . . 4 ⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)
161	119, 160	eleqtrrd 2916	. . 3 ⊢ (𝜑 → (𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))
162		fvex 6677	. . . 4 ⊢ (𝐻‘(card‘𝑍)) ∈ V
163	162	eliniseg 5952	. . . 4 ⊢ ((𝐻‘(card‘𝑍)) ∈ V → ((𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍))))
164	162, 163	ax-mp 5	. . 3 ⊢ ((𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
165	161, 164	sylib 220	. 2 ⊢ (𝜑 → (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
166	26	simprd 498	. . . . 5 ⊢ (𝜑 → ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))
167	166	simpld 497	. . . 4 ⊢ (𝜑 → (𝑊‘𝑍) We 𝑍)
168		weso 5540	. . . 4 ⊢ ((𝑊‘𝑍) We 𝑍 → (𝑊‘𝑍) Or 𝑍)
169	167, 168	syl 17	. . 3 ⊢ (𝜑 → (𝑊‘𝑍) Or 𝑍)
170		sonr 5490	. . 3 ⊢ (((𝑊‘𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
171	169, 119, 170	syl2anc 586	. 2 ⊢ (𝜑 → ¬ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))
172	165, 171	pm2.65i 196	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3494  [wsbc 3771   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935   ⊊ wpss 3936  ifcif 4466  𝒫 cpw 4538  {csn 4560  ∪ cuni 4831  ∩ cint 4868  ∪ ciun 4911   class class class wbr 5058  {copab 5120   Or wor 5467   We wwe 5507   × cxp 5547  ^◡ccnv 5548  dom cdm 5549  ran crn 5550   “ cima 5552  Oncon0 6185  –1-1→wf1 6346  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  ωcom 7574   ↑_m cmap 8400   ≈ cen 8500   ≼ cdom 8501   ≺ csdm 8502  Fincfn 8503  cardccrd 9358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-oi 8968  df-card 9362

This theorem is referenced by:  pwfseqlem5  10079