Theorem pwfseqlem1 10080

Description: Lemma for pwfseq 10086. 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
4		f1f 6575	. . . . 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 4056	. . . . . 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 1214	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) → 𝑥 ⊆ 𝐴)
9	7, 8	sylan2b 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ⊆ 𝐴)
10	6, 9	sstrid 3978	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ⊆ 𝐴)
11		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
12	11	rabex 5235	. . . . . 6 ⊢ {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ V
13	12	elpw 4543	. . . . 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 236	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → {𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))} ∈ 𝒫 𝐴)
15	5, 14	ffvelrnd 6852	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
16	1, 15	eqeltrid 2917	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
17		pm5.19 390	. . 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 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
20		f1f 6575	. . . . . . . 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		ffvelrn 6849	. . . . . . 7 ⊢ ((𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)⟶𝑥 ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (𝐾‘𝐷) ∈ 𝑥)
23	21, 22	sylancom 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (𝐾‘𝐷) ∈ 𝑥)
24		f1f1orn 6626	. . . . . . . . 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 7033	. . . . . . . 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 590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐾‘(𝐾‘𝐷)) = 𝐷)
28		f1fn 6576	. . . . . . . . . . 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 6848	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) ∈ ran 𝐺)
32	1, 31	eqeltrid 2917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ran 𝐺)
33	32	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → 𝐷 ∈ ran 𝐺)
34	27, 33	eqeltrd 2913	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) → (◡𝐾‘(𝐾‘𝐷)) ∈ ran 𝐺)
35		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐾‘𝐷) → (◡𝐾‘𝑦) = (◡𝐾‘(𝐾‘𝐷)))
36	35	eleq1d 2897	. . . . . . . . . 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 6675	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐾‘𝐷) → (◡𝐺‘(◡𝐾‘𝑦)) = (◡𝐺‘(◡𝐾‘(𝐾‘𝐷))))
39	37, 38	eleq12d 2907	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐾‘𝐷) → (𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦)) ↔ (𝐾‘𝐷) ∈ (◡𝐺‘(◡𝐾‘(𝐾‘𝐷)))))
40	39	notbid 320	. . . . . . . . . 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 6670	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (◡𝐾‘𝑤) = (◡𝐾‘𝑦))
43	42	eleq1d 2897	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ((◡𝐾‘𝑤) ∈ ran 𝐺 ↔ (◡𝐾‘𝑦) ∈ ran 𝐺))
44		id 22	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → 𝑤 = 𝑦)
45		2fveq3 6675	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (◡𝐺‘(◡𝐾‘𝑤)) = (◡𝐺‘(◡𝐾‘𝑦)))
46	44, 45	eleq12d 2907	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)) ↔ 𝑦 ∈ (◡𝐺‘(◡𝐾‘𝑦))))
47	46	notbid 320	. . . . . . . . . . 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 3491	. . . . . . . . 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 3683	. . . . . . . 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 471	. . . . . . . 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 280	. . . . . . 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 538	. . . . . 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 586	. . . . 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	syl6eq 2872	. . . . . . . . 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 6674	. . . . . . . 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 6626	. . . . . . . . . . 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 7033	. . . . . . . . . 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 586	. . . . . . . . 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 483	. . . . . . . 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 2856	. . . . . . 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 2898	. . . . . 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 320	. . . . 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 281	. . . 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 415	. . 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 201	. 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))
68	16, 67	eldifd 3947	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3142   ∖ cdif 3933   ⊆ wss 3936  𝒫 cpw 4539  ∪ ciun 4919   class class class wbr 5066   We wwe 5513   × cxp 5553  ^◡ccnv 5554  ran crn 5556   Fn wfn 6350  ⟶wf 6351  –1-1→wf1 6352  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ↑_m cmap 8406   ≼ cdom 8507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363

This theorem is referenced by:  pwfseqlem3  10082