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Theorem fin1a2lem13 9178

Description: Lemma for fin1a2 9181. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
fin1a2lem13	⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∖ 𝐶) ∈ Fin^II)

Proof of Theorem fin1a2lem13 Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → (𝐵 ∖ 𝐶) ∈ Fin^II)
2		simpll1 1098	. . . 4 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → 𝐴 ⊆ 𝒫 𝐵)
3		ssel2 3578	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝒫 𝐵 ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ 𝒫 𝐵)
4	3	elpwid 4141	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝒫 𝐵 ∧ 𝑔 ∈ 𝐴) → 𝑔 ⊆ 𝐵)
5	4	ssdifd 3724	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝒫 𝐵 ∧ 𝑔 ∈ 𝐴) → (𝑔 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
6		sseq1 3605	. . . . . . . 8 ⊢ (𝑓 = (𝑔 ∖ 𝐶) → (𝑓 ⊆ (𝐵 ∖ 𝐶) ↔ (𝑔 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)))
7	5, 6	syl5ibrcom 237	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝒫 𝐵 ∧ 𝑔 ∈ 𝐴) → (𝑓 = (𝑔 ∖ 𝐶) → 𝑓 ⊆ (𝐵 ∖ 𝐶)))
8	7	rexlimdva 3024	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐵 → (∃𝑔 ∈ 𝐴 𝑓 = (𝑔 ∖ 𝐶) → 𝑓 ⊆ (𝐵 ∖ 𝐶)))
9		vex 3189	. . . . . . 7 ⊢ 𝑓 ∈ V
10		eqid 2621	. . . . . . . 8 ⊢ (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))
11	10	elrnmpt 5332	. . . . . . 7 ⊢ (𝑓 ∈ V → (𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 𝑓 = (𝑔 ∖ 𝐶)))
12	9, 11	ax-mp 5	. . . . . 6 ⊢ (𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 𝑓 = (𝑔 ∖ 𝐶))
13		selpw 4137	. . . . . 6 ⊢ (𝑓 ∈ 𝒫 (𝐵 ∖ 𝐶) ↔ 𝑓 ⊆ (𝐵 ∖ 𝐶))
14	8, 12, 13	3imtr4g 285	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐵 → (𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → 𝑓 ∈ 𝒫 (𝐵 ∖ 𝐶)))
15	14	ssrdv 3589	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ⊆ 𝒫 (𝐵 ∖ 𝐶))
16	2, 15	syl 17	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ⊆ 𝒫 (𝐵 ∖ 𝐶))
17		simplrr 800	. . . 4 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → 𝐶 ∈ 𝐴)
18		difid 3922	. . . . . . 7 ⊢ (𝐶 ∖ 𝐶) = ∅
19	18	eqcomi 2630	. . . . . 6 ⊢ ∅ = (𝐶 ∖ 𝐶)
20		difeq1 3699	. . . . . . . 8 ⊢ (𝑔 = 𝐶 → (𝑔 ∖ 𝐶) = (𝐶 ∖ 𝐶))
21	20	eqeq2d 2631	. . . . . . 7 ⊢ (𝑔 = 𝐶 → (∅ = (𝑔 ∖ 𝐶) ↔ ∅ = (𝐶 ∖ 𝐶)))
22	21	rspcev 3295	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∅ = (𝐶 ∖ 𝐶)) → ∃𝑔 ∈ 𝐴 ∅ = (𝑔 ∖ 𝐶))
23	19, 22	mpan2 706	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → ∃𝑔 ∈ 𝐴 ∅ = (𝑔 ∖ 𝐶))
24		0ex 4750	. . . . . 6 ⊢ ∅ ∈ V
25	10	elrnmpt 5332	. . . . . 6 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 ∅ = (𝑔 ∖ 𝐶)))
26	24, 25	ax-mp 5	. . . . 5 ⊢ (∅ ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 ∅ = (𝑔 ∖ 𝐶))
27	23, 26	sylibr 224	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ∅ ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
28		ne0i 3897	. . . 4 ⊢ (∅ ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ≠ ∅)
29	17, 27, 28	3syl 18	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ≠ ∅)
30		simpll2 1099	. . . 4 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → [⊊] Or 𝐴)
31		vex 3189	. . . . . . . 8 ⊢ 𝑥 ∈ V
32	10	elrnmpt 5332	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 𝑥 = (𝑔 ∖ 𝐶)))
33	31, 32	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 𝑥 = (𝑔 ∖ 𝐶))
34		difeq1 3699	. . . . . . . . . 10 ⊢ (𝑔 = 𝑒 → (𝑔 ∖ 𝐶) = (𝑒 ∖ 𝐶))
35	34	eqeq2d 2631	. . . . . . . . 9 ⊢ (𝑔 = 𝑒 → (𝑥 = (𝑔 ∖ 𝐶) ↔ 𝑥 = (𝑒 ∖ 𝐶)))
36	35	cbvrexv 3160	. . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐴 𝑥 = (𝑔 ∖ 𝐶) ↔ ∃𝑒 ∈ 𝐴 𝑥 = (𝑒 ∖ 𝐶))
37		sorpssi 6896	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑒 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴)) → (𝑒 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑒))
38		ssdif 3723	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 ⊆ 𝑔 → (𝑒 ∖ 𝐶) ⊆ (𝑔 ∖ 𝐶))
39		ssdif 3723	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ⊆ 𝑒 → (𝑔 ∖ 𝐶) ⊆ (𝑒 ∖ 𝐶))
40	38, 39	orim12i 538	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑒) → ((𝑒 ∖ 𝐶) ⊆ (𝑔 ∖ 𝐶) ∨ (𝑔 ∖ 𝐶) ⊆ (𝑒 ∖ 𝐶)))
41	37, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (( [⊊] Or 𝐴 ∧ (𝑒 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴)) → ((𝑒 ∖ 𝐶) ⊆ (𝑔 ∖ 𝐶) ∨ (𝑔 ∖ 𝐶) ⊆ (𝑒 ∖ 𝐶)))
42		sseq2 3606	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑔 ∖ 𝐶) → ((𝑒 ∖ 𝐶) ⊆ 𝑓 ↔ (𝑒 ∖ 𝐶) ⊆ (𝑔 ∖ 𝐶)))
43		sseq1 3605	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑔 ∖ 𝐶) → (𝑓 ⊆ (𝑒 ∖ 𝐶) ↔ (𝑔 ∖ 𝐶) ⊆ (𝑒 ∖ 𝐶)))
44	42, 43	orbi12d 745	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑔 ∖ 𝐶) → (((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶)) ↔ ((𝑒 ∖ 𝐶) ⊆ (𝑔 ∖ 𝐶) ∨ (𝑔 ∖ 𝐶) ⊆ (𝑒 ∖ 𝐶))))
45	41, 44	syl5ibrcom 237	. . . . . . . . . . . . . 14 ⊢ (( [⊊] Or 𝐴 ∧ (𝑒 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴)) → (𝑓 = (𝑔 ∖ 𝐶) → ((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶))))
46	45	expr 642	. . . . . . . . . . . . 13 ⊢ (( [⊊] Or 𝐴 ∧ 𝑒 ∈ 𝐴) → (𝑔 ∈ 𝐴 → (𝑓 = (𝑔 ∖ 𝐶) → ((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶)))))
47	46	rexlimdv 3023	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝑒 ∈ 𝐴) → (∃𝑔 ∈ 𝐴 𝑓 = (𝑔 ∖ 𝐶) → ((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶))))
48	12, 47	syl5bi 232	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ 𝑒 ∈ 𝐴) → (𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → ((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶))))
49	48	ralrimiv 2959	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝑒 ∈ 𝐴) → ∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶)))
50		sseq1 3605	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑒 ∖ 𝐶) → (𝑥 ⊆ 𝑓 ↔ (𝑒 ∖ 𝐶) ⊆ 𝑓))
51		sseq2 3606	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑒 ∖ 𝐶) → (𝑓 ⊆ 𝑥 ↔ 𝑓 ⊆ (𝑒 ∖ 𝐶)))
52	50, 51	orbi12d 745	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑒 ∖ 𝐶) → ((𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥) ↔ ((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶))))
53	52	ralbidv 2980	. . . . . . . . . 10 ⊢ (𝑥 = (𝑒 ∖ 𝐶) → (∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥) ↔ ∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))((𝑒 ∖ 𝐶) ⊆ 𝑓 ∨ 𝑓 ⊆ (𝑒 ∖ 𝐶))))
54	49, 53	syl5ibrcom 237	. . . . . . . . 9 ⊢ (( [⊊] Or 𝐴 ∧ 𝑒 ∈ 𝐴) → (𝑥 = (𝑒 ∖ 𝐶) → ∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥)))
55	54	rexlimdva 3024	. . . . . . . 8 ⊢ ( [⊊] Or 𝐴 → (∃𝑒 ∈ 𝐴 𝑥 = (𝑒 ∖ 𝐶) → ∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥)))
56	36, 55	syl5bi 232	. . . . . . 7 ⊢ ( [⊊] Or 𝐴 → (∃𝑔 ∈ 𝐴 𝑥 = (𝑔 ∖ 𝐶) → ∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥)))
57	33, 56	syl5bi 232	. . . . . 6 ⊢ ( [⊊] Or 𝐴 → (𝑥 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → ∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥)))
58	57	ralrimiv 2959	. . . . 5 ⊢ ( [⊊] Or 𝐴 → ∀𝑥 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥))
59		sorpss 6895	. . . . 5 ⊢ ( [⊊] Or ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∀𝑥 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))∀𝑓 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))(𝑥 ⊆ 𝑓 ∨ 𝑓 ⊆ 𝑥))
60	58, 59	sylibr 224	. . . 4 ⊢ ( [⊊] Or 𝐴 → [⊊] Or ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
61	30, 60	syl 17	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → [⊊] Or ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
62		fin2i 9061	. . 3 ⊢ ((((𝐵 ∖ 𝐶) ∈ Fin^II ∧ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ⊆ 𝒫 (𝐵 ∖ 𝐶)) ∧ (ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ≠ ∅ ∧ [⊊] Or ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))) → ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
63	1, 16, 29, 61, 62	syl22anc 1324	. 2 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
64		simpll3 1100	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → ¬ ∪ 𝐴 ∈ 𝐴)
65		difeq1 3699	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 ∖ 𝐶) = (𝑓 ∖ 𝐶))
66	65	cbvmptv 4710	. . . . . 6 ⊢ (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∈ 𝐴 ↦ (𝑓 ∖ 𝐶))
67	66	elrnmpt 5332	. . . . 5 ⊢ (∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → (∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑓 ∈ 𝐴 ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)))
68	67	ibi 256	. . . 4 ⊢ (∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → ∃𝑓 ∈ 𝐴 ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))
69		eqid 2621	. . . . . . . . . . . . . . . 16 ⊢ (ℎ ∖ 𝐶) = (ℎ ∖ 𝐶)
70		difeq1 3699	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = ℎ → (𝑔 ∖ 𝐶) = (ℎ ∖ 𝐶))
71	70	eqeq2d 2631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → ((ℎ ∖ 𝐶) = (𝑔 ∖ 𝐶) ↔ (ℎ ∖ 𝐶) = (ℎ ∖ 𝐶)))
72	71	rspcev 3295	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ ∈ 𝐴 ∧ (ℎ ∖ 𝐶) = (ℎ ∖ 𝐶)) → ∃𝑔 ∈ 𝐴 (ℎ ∖ 𝐶) = (𝑔 ∖ 𝐶))
73	69, 72	mpan2 706	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ 𝐴 → ∃𝑔 ∈ 𝐴 (ℎ ∖ 𝐶) = (𝑔 ∖ 𝐶))
74	73	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ ℎ ∈ 𝐴) → ∃𝑔 ∈ 𝐴 (ℎ ∖ 𝐶) = (𝑔 ∖ 𝐶))
75		vex 3189	. . . . . . . . . . . . . . 15 ⊢ ℎ ∈ V
76		difexg 4768	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ V → (ℎ ∖ 𝐶) ∈ V)
77	10	elrnmpt 5332	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ∖ 𝐶) ∈ V → ((ℎ ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 (ℎ ∖ 𝐶) = (𝑔 ∖ 𝐶)))
78	75, 76, 77	mp2b 10	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 (ℎ ∖ 𝐶) = (𝑔 ∖ 𝐶))
79	74, 78	sylibr 224	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ ℎ ∈ 𝐴) → (ℎ ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
80		elssuni 4433	. . . . . . . . . . . . 13 ⊢ ((ℎ ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → (ℎ ∖ 𝐶) ⊆ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
81	79, 80	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ ℎ ∈ 𝐴) → (ℎ ∖ 𝐶) ⊆ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
82		simplr 791	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ ℎ ∈ 𝐴) → ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))
83	81, 82	sseqtrd 3620	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ ℎ ∈ 𝐴) → (ℎ ∖ 𝐶) ⊆ (𝑓 ∖ 𝐶))
84	83	adantll 749	. . . . . . . . . 10 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → (ℎ ∖ 𝐶) ⊆ (𝑓 ∖ 𝐶))
85		unss2 3762	. . . . . . . . . . 11 ⊢ ((ℎ ∖ 𝐶) ⊆ (𝑓 ∖ 𝐶) → (𝐶 ∪ (ℎ ∖ 𝐶)) ⊆ (𝐶 ∪ (𝑓 ∖ 𝐶)))
86		uncom 3735	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∪ (ℎ ∖ 𝐶)) = ((ℎ ∖ 𝐶) ∪ 𝐶)
87		undif1 4015	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ∖ 𝐶) ∪ 𝐶) = (ℎ ∪ 𝐶)
88	86, 87	eqtri 2643	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∪ (ℎ ∖ 𝐶)) = (ℎ ∪ 𝐶)
89	88	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → (𝐶 ∪ (ℎ ∖ 𝐶)) = (ℎ ∪ 𝐶))
90	64	ad2antrr 761	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ¬ ∪ 𝐴 ∈ 𝐴)
91	17	ad2antrr 761	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → 𝐶 ∈ 𝐴)
92		simplrr 800	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))
93		eqid 2621	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∖ 𝐶) = (𝑥 ∖ 𝐶)
94		difeq1 3699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑔 = 𝑥 → (𝑔 ∖ 𝐶) = (𝑥 ∖ 𝐶))
95	94	eqeq2d 2631	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑔 = 𝑥 → ((𝑥 ∖ 𝐶) = (𝑔 ∖ 𝐶) ↔ (𝑥 ∖ 𝐶) = (𝑥 ∖ 𝐶)))
96	95	rspcev 3295	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∖ 𝐶) = (𝑥 ∖ 𝐶)) → ∃𝑔 ∈ 𝐴 (𝑥 ∖ 𝐶) = (𝑔 ∖ 𝐶))
97	93, 96	mpan2 706	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ 𝐴 → ∃𝑔 ∈ 𝐴 (𝑥 ∖ 𝐶) = (𝑔 ∖ 𝐶))
98		difexg 4768	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ V → (𝑥 ∖ 𝐶) ∈ V)
99	10	elrnmpt 5332	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∖ 𝐶) ∈ V → ((𝑥 ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 (𝑥 ∖ 𝐶) = (𝑔 ∖ 𝐶)))
100	31, 98, 99	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ↔ ∃𝑔 ∈ 𝐴 (𝑥 ∖ 𝐶) = (𝑔 ∖ 𝐶))
101	97, 100	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
102	101	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
103		simpllr 798	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))
104		ssdif0 3916	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 ⊆ 𝐶 ↔ (𝑓 ∖ 𝐶) = ∅)
105	104	biimpi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ⊆ 𝐶 → (𝑓 ∖ 𝐶) = ∅)
106	105	ad2antlr 762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝑓 ∖ 𝐶) = ∅)
107	103, 106	eqtrd 2655	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = ∅)
108		uni0c 4430	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = ∅ ↔ ∀𝑒 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))𝑒 = ∅)
109	107, 108	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ∀𝑒 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))𝑒 = ∅)
110		eqeq1 2625	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = (𝑥 ∖ 𝐶) → (𝑒 = ∅ ↔ (𝑥 ∖ 𝐶) = ∅))
111	110	rspcva 3293	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∖ 𝐶) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∧ ∀𝑒 ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶))𝑒 = ∅) → (𝑥 ∖ 𝐶) = ∅)
112	102, 109, 111	syl2anc 692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∖ 𝐶) = ∅)
113		ssdif0 3916	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ⊆ 𝐶 ↔ (𝑥 ∖ 𝐶) = ∅)
114	112, 113	sylibr 224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐶)
115	114	ralrimiva 2960	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐶)
116		unissb 4435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ 𝐴 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐶)
117	115, 116	sylibr 224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) → ∪ 𝐴 ⊆ 𝐶)
118		elssuni 4433	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ⊆ ∪ 𝐴)
119	118	ad2antrr 761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) → 𝐶 ⊆ ∪ 𝐴)
120	117, 119	eqssd 3600	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) → ∪ 𝐴 = 𝐶)
121		simpll 789	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) → 𝐶 ∈ 𝐴)
122	120, 121	eqeltrd 2698	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) ∧ 𝑓 ⊆ 𝐶) → ∪ 𝐴 ∈ 𝐴)
123	122	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶)) → (𝑓 ⊆ 𝐶 → ∪ 𝐴 ∈ 𝐴))
124	91, 92, 123	syl2anc 692	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → (𝑓 ⊆ 𝐶 → ∪ 𝐴 ∈ 𝐴))
125	90, 124	mtod 189	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ¬ 𝑓 ⊆ 𝐶)
126	30	ad2antrr 761	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → [⊊] Or 𝐴)
127		simplrl 799	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → 𝑓 ∈ 𝐴)
128		sorpssi 6896	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑓 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑓 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝑓))
129	126, 127, 91, 128	syl12anc 1321	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → (𝑓 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝑓))
130		orel1 397	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑓 ⊆ 𝐶 → ((𝑓 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝑓) → 𝐶 ⊆ 𝑓))
131	125, 129, 130	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → 𝐶 ⊆ 𝑓)
132		undif 4021	. . . . . . . . . . . . . 14 ⊢ (𝐶 ⊆ 𝑓 ↔ (𝐶 ∪ (𝑓 ∖ 𝐶)) = 𝑓)
133	131, 132	sylib 208	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → (𝐶 ∪ (𝑓 ∖ 𝐶)) = 𝑓)
134	89, 133	sseq12d 3613	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ((𝐶 ∪ (ℎ ∖ 𝐶)) ⊆ (𝐶 ∪ (𝑓 ∖ 𝐶)) ↔ (ℎ ∪ 𝐶) ⊆ 𝑓))
135		ssun1 3754	. . . . . . . . . . . . 13 ⊢ ℎ ⊆ (ℎ ∪ 𝐶)
136		sstr 3591	. . . . . . . . . . . . 13 ⊢ ((ℎ ⊆ (ℎ ∪ 𝐶) ∧ (ℎ ∪ 𝐶) ⊆ 𝑓) → ℎ ⊆ 𝑓)
137	135, 136	mpan 705	. . . . . . . . . . . 12 ⊢ ((ℎ ∪ 𝐶) ⊆ 𝑓 → ℎ ⊆ 𝑓)
138	134, 137	syl6bi 243	. . . . . . . . . . 11 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ((𝐶 ∪ (ℎ ∖ 𝐶)) ⊆ (𝐶 ∪ (𝑓 ∖ 𝐶)) → ℎ ⊆ 𝑓))
139	85, 138	syl5 34	. . . . . . . . . 10 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ((ℎ ∖ 𝐶) ⊆ (𝑓 ∖ 𝐶) → ℎ ⊆ 𝑓))
140	84, 139	mpd 15	. . . . . . . . 9 ⊢ ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) ∧ ℎ ∈ 𝐴) → ℎ ⊆ 𝑓)
141	140	ralrimiva 2960	. . . . . . . 8 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) → ∀ℎ ∈ 𝐴 ℎ ⊆ 𝑓)
142		unissb 4435	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑓 ↔ ∀ℎ ∈ 𝐴 ℎ ⊆ 𝑓)
143	141, 142	sylibr 224	. . . . . . 7 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) → ∪ 𝐴 ⊆ 𝑓)
144		elssuni 4433	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ ∪ 𝐴)
145	144	ad2antrl 763	. . . . . . 7 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) → 𝑓 ⊆ ∪ 𝐴)
146	143, 145	eqssd 3600	. . . . . 6 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) → ∪ 𝐴 = 𝑓)
147		simprl 793	. . . . . 6 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) → 𝑓 ∈ 𝐴)
148	146, 147	eqeltrd 2698	. . . . 5 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) ∧ (𝑓 ∈ 𝐴 ∧ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶))) → ∪ 𝐴 ∈ 𝐴)
149	148	rexlimdvaa 3025	. . . 4 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → (∃𝑓 ∈ 𝐴 ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) = (𝑓 ∖ 𝐶) → ∪ 𝐴 ∈ 𝐴))
150	68, 149	syl5 34	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → (∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) → ∪ 𝐴 ∈ 𝐴))
151	64, 150	mtod 189	. 2 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) ∧ (𝐵 ∖ 𝐶) ∈ Fin^II) → ¬ ∪ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)) ∈ ran (𝑔 ∈ 𝐴 ↦ (𝑔 ∖ 𝐶)))
152	63, 151	pm2.65da 599	1 ⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∖ 𝐶) ∈ Fin^II)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1987 ≠ wne 2790 ∀wral 2907 ∃wrex 2908 Vcvv 3186 ∖ cdif 3552 ∪ cun 3553 ⊆ wss 3555 ∅c0 3891 𝒫 cpw 4130 ∪ cuni 4402 ↦ cmpt 4673 Or wor 4994 ran crn 5075 [⊊] crpss 6889 Fincfn 7899 Fin^IIcfin2 9045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-sep 4741 ax-nul 4749 ax-pow 4803 ax-pr 4867

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-ral 2912 df-rex 2913 df-rab 2916 df-v 3188 df-dif 3558 df-un 3560 df-in 3562 df-ss 3569 df-pss 3571 df-nul 3892 df-if 4059 df-pw 4132 df-sn 4149 df-pr 4151 df-op 4155 df-uni 4403 df-br 4614 df-opab 4674 df-mpt 4675 df-po 4995 df-so 4996 df-xp 5080 df-rel 5081 df-cnv 5082 df-dm 5084 df-rn 5085 df-rpss 6890 df-fin2 9052

This theorem is referenced by: fin1a2s 9180

W3C validator