Theorem pwfseqlem1 10698

Description: Lemma for pwfseq 10704. Derive a contradiction by diagonalization. (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)))})

Assertion

Ref	Expression
pwfseqlem1	⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))

Distinct variable groups:   𝑛,𝑟,𝑤,𝑥   𝐷,𝑛   𝑤,𝐺   𝑤,𝐾   𝐻,𝑟,𝑥   𝜑,𝑛,𝑟,𝑥   𝜓,𝑛   𝐴,𝑛,𝑟,𝑥

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑟)   𝐴(𝑤)   𝐷(𝑥,𝑤,𝑟)   𝐺(𝑥,𝑛,𝑟)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑛,𝑟)   𝑋(𝑥,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwfseqlem4.d	. . 3 ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
2		pwfseqlem4.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
3	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
4		f1f 6804	. . . . 5 ⊢ (𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → 𝐺:𝒫 𝐴⟶∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
5	3, 4	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴⟶∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
6		ssrab2 4080	. . . . . 6 ⊢ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ⊆ 𝑥
7		pwfseqlem4.ps	. . . . . . 7 ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
8		simprl1 1219	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) → 𝑥 ⊆ 𝐴)
9	7, 8	sylan2b 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ⊆ 𝐴)
10	6, 9	sstrid 3995	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ⊆ 𝐴)
11		vex 3484	. . . . . . 7 ⊢ 𝑥 ∈ V
12	11	rabex 5339	. . . . . 6 ⊢ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ V
13	12	elpw 4604	. . . . 5 ⊢ ({𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ 𝒫 𝐴 ↔ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ⊆ 𝐴)
14	10, 13	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ 𝒫 𝐴)
15	5, 14	ffvelcdmd 7105	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
16	1, 15	eqeltrid 2845	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
17		pm5.19 386	. . 3 ⊢ ¬ ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ ¬ (𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
18		pwfseqlem4.k	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
20		f1f 6804	. . . . . . . 8 ⊢ (𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥 → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)⟶𝑥)
21	19, 20	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)⟶𝑥)
22		ffvelcdm 7101	. . . . . . 7 ⊢ ((𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)⟶𝑥 ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (𝐾‘𝐷) ∈ 𝑥)
23	21, 22	sylancom 588	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (𝐾‘𝐷) ∈ 𝑥)
24		f1f1orn 6859	. . . . . . . . 9 ⊢ (𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥 → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1-onto→ran 𝐾)
25	19, 24	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1-onto→ran 𝐾)
26		f1ocnvfv1 7296	. . . . . . . 8 ⊢ ((𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1-onto→ran 𝐾 ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐾‘(𝐾‘𝐷)) = 𝐷)
27	25, 26	sylancom 588	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐾‘(𝐾‘𝐷)) = 𝐷)
28		f1fn 6805	. . . . . . . . . . 11 ⊢ (𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → 𝐺 Fn 𝒫 𝐴)
29	3, 28	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐺 Fn 𝒫 𝐴)
30		fnfvelrn 7100	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝒫 𝐴 ∧ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ 𝒫 𝐴) → (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) ∈ ran 𝐺)
31	29, 14, 30	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) ∈ ran 𝐺)
32	1, 31	eqeltrid 2845	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ran 𝐺)
33	32	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → 𝐷 ∈ ran 𝐺)
34	27, 33	eqeltrd 2841	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺)
35		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐾‘𝐷) → (◡𝐾‘𝑦) = (◡𝐾‘(𝐾‘𝐷)))
36	35	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑦 = (𝐾‘𝐷) → ((◡𝐾‘𝑦) ∈ ran 𝐺 ↔ (◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺))
37		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐾‘𝐷) → 𝑦 = (𝐾‘𝐷))
38		2fveq3 6911	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐾‘𝐷) → (◡𝐺‘(◡𝐾‘𝑦)) = (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))))
39	37, 38	eleq12d 2835	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐾‘𝐷) → (𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦)) ↔ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))))
40	39	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = (𝐾‘𝐷) → (¬ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦)) ↔ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))))
41	36, 40	anbi12d 632	. . . . . . . . 9 ⊢ (𝑦 = (𝐾‘𝐷) → (((◡𝐾‘𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦))) ↔ ((◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))))))
42		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (◡𝐾‘𝑤) = (◡𝐾‘𝑦))
43	42	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ((◡𝐾‘𝑤) ∈ ran 𝐺 ↔ (◡𝐾‘𝑦) ∈ ran 𝐺))
44		id 22	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → 𝑤 = 𝑦)
45		2fveq3 6911	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (◡𝐺‘(◡𝐾‘𝑤)) = (◡𝐺‘(◡𝐾‘𝑦)))
46	44, 45	eleq12d 2835	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)) ↔ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦))))
47	46	notbid 318	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)) ↔ ¬ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦))))
48	43, 47	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤))) ↔ ((◡𝐾‘𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦)))))
49	48	cbvrabv 3447	. . . . . . . . 9 ⊢ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} = {𝑦 ∈ 𝑥 ∣ ((◡𝐾‘𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦)))}
50	41, 49	elrab2 3695	. . . . . . . 8 ⊢ ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ ((𝐾‘𝐷) ∈ 𝑥 ∧ ((◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))))))
51		anass 468	. . . . . . . 8 ⊢ ((((𝐾‘𝐷) ∈ 𝑥 ∧ (◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺) ∧ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))) ↔ ((𝐾‘𝐷) ∈ 𝑥 ∧ ((◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))))))
52	50, 51	bitr4i 278	. . . . . . 7 ⊢ ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ (((𝐾‘𝐷) ∈ 𝑥 ∧ (◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺) ∧ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))))
53	52	baib 535	. . . . . 6 ⊢ (((𝐾‘𝐷) ∈ 𝑥 ∧ (◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺) → ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))))
54	23, 34, 53	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ ¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))))
55	27, 1	eqtrdi 2793	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐾‘(𝐾‘𝐷)) = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}))
56	55	fveq2d 6910	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))) = (◡𝐺‘(𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})))
57		f1f1orn 6859	. . . . . . . . . . 11 ⊢ (𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → 𝐺:𝒫 𝐴–1-1-onto→ran 𝐺)
58	3, 57	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴–1-1-onto→ran 𝐺)
59		f1ocnvfv1 7296	. . . . . . . . . 10 ⊢ ((𝐺:𝒫 𝐴–1-1-onto→ran 𝐺 ∧ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ 𝒫 𝐴) → (◡𝐺‘(𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})) = {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
60	58, 14, 59	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (◡𝐺‘(𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})) = {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
61	60	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐺‘(𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})) = {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
62	56, 61	eqtrd 2777	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))) = {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
63	62	eleq2d 2827	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → ((𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))) ↔ (𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}))
64	63	notbid 318	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (¬ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))) ↔ ¬ (𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}))
65	54, 64	bitrd 279	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ ¬ (𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}))
66	65	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛) → ((𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ↔ ¬ (𝐾‘𝐷) ∈ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})))
67	17, 66	mtoi 199	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))
68	16, 67	eldifd 3962	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {crab 3436   ∖ cdif 3948   ⊆ wss 3951  𝒫 cpw 4600  ∪ ciun 4991   class class class wbr 5143   We wwe 5636   × cxp 5683  ^◡ccnv 5684  ran crn 5686   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  ωcom 7887   ↑_m cmap 8866   ≼ cdom 8983

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569

This theorem is referenced by:  pwfseqlem3  10700