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Theorem fin1a2lem12 9822

Description: Lemma for fin1a2 9826. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
fin1a2lem12	⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ Fin^III)

Proof of Theorem fin1a2lem12 Dummy variables 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 488	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐵 ∈ Fin^III)
2		simpll1 1209	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ⊆ 𝒫 𝐵)
3	2	adantr 484	. . . . . 6 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑒 ∈ ω) → 𝐴 ⊆ 𝒫 𝐵)
4		ssrab2 4007	. . . . . . . 8 ⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐴
5	4	unissi 4809	. . . . . . 7 ⊢ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ ∪ 𝐴
6		sspwuni 4985	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
7	6	biimpi 219	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)
8	5, 7	sstrid 3926	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵)
9	3, 8	syl 17	. . . . 5 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑒 ∈ ω) → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵)
10		elpw2g 5211	. . . . . 6 ⊢ (𝐵 ∈ Fin^III → (∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵 ↔ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵))
11	10	ad2antlr 726	. . . . 5 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑒 ∈ ω) → (∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵 ↔ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵))
12	9, 11	mpbird 260	. . . 4 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑒 ∈ ω) → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵)
13	12	fmpttd 6856	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}):ω⟶𝒫 𝐵)
14		vex 3444	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
15	14	sucex 7506	. . . . . . . . . 10 ⊢ suc 𝑑 ∈ V
16		sssucid 6236	. . . . . . . . . 10 ⊢ 𝑑 ⊆ suc 𝑑
17		ssdomg 8538	. . . . . . . . . 10 ⊢ (suc 𝑑 ∈ V → (𝑑 ⊆ suc 𝑑 → 𝑑 ≼ suc 𝑑))
18	15, 16, 17	mp2 9	. . . . . . . . 9 ⊢ 𝑑 ≼ suc 𝑑
19		domtr 8545	. . . . . . . . 9 ⊢ ((𝑓 ≼ 𝑑 ∧ 𝑑 ≼ suc 𝑑) → 𝑓 ≼ suc 𝑑)
20	18, 19	mpan2 690	. . . . . . . 8 ⊢ (𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑)
21	20	a1i 11	. . . . . . 7 ⊢ (𝑓 ∈ 𝐴 → (𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑))
22	21	ss2rabi 4004	. . . . . 6 ⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}
23		uniss 4808	. . . . . 6 ⊢ ({𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
24	22, 23	mp1i 13	. . . . 5 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑑 ∈ ω) → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
25		id 22	. . . . . 6 ⊢ (𝑑 ∈ ω → 𝑑 ∈ ω)
26		pwexg 5244	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin^III → 𝒫 𝐵 ∈ V)
27	26	adantl 485	. . . . . . . 8 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝒫 𝐵 ∈ V)
28	27, 2	ssexd 5192	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ∈ V)
29		rabexg 5198	. . . . . . 7 ⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V)
30		uniexg 7446	. . . . . . 7 ⊢ ({𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V)
31	28, 29, 30	3syl 18	. . . . . 6 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V)
32		breq2 5034	. . . . . . . . 9 ⊢ (𝑒 = 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ 𝑑))
33	32	rabbidv 3427	. . . . . . . 8 ⊢ (𝑒 = 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
34	33	unieqd 4814	. . . . . . 7 ⊢ (𝑒 = 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
35		eqid 2798	. . . . . . 7 ⊢ (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})
36	34, 35	fvmptg 6743	. . . . . 6 ⊢ ((𝑑 ∈ ω ∧ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
37	25, 31, 36	syl2anr 599	. . . . 5 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
38		peano2 7582	. . . . . 6 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
39		rabexg 5198	. . . . . . 7 ⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V)
40		uniexg 7446	. . . . . . 7 ⊢ ({𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V)
41	28, 39, 40	3syl 18	. . . . . 6 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V)
42		breq2 5034	. . . . . . . . 9 ⊢ (𝑒 = suc 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ suc 𝑑))
43	42	rabbidv 3427	. . . . . . . 8 ⊢ (𝑒 = suc 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
44	43	unieqd 4814	. . . . . . 7 ⊢ (𝑒 = suc 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
45	44, 35	fvmptg 6743	. . . . . 6 ⊢ ((suc 𝑑 ∈ ω ∧ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
46	38, 41, 45	syl2anr 599	. . . . 5 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
47	24, 37, 46	3sstr4d 3962	. . . 4 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑))
48	47	ralrimiva 3149	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ∀𝑑 ∈ ω ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑))
49		fin34i 9792	. . 3 ⊢ ((𝐵 ∈ Fin^III ∧ (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}):ω⟶𝒫 𝐵 ∧ ∀𝑑 ∈ ω ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑)) → ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}))
50	1, 13, 48, 49	syl3anc 1368	. 2 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}))
51		fin1a2lem11 9821	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
52	51	adantrr 716	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
53	52	3ad2antl2 1183	. . . 4 ⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
54	53	adantr 484	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
55		simpll3 1211	. . . . . 6 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ ∪ 𝐴 ∈ 𝐴)
56		simplrr 777	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ≠ ∅)
57		sspwuni 4985	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 ⊆ ∅)
58		ss0b 4305	. . . . . . . . . . 11 ⊢ (∪ 𝐴 ⊆ ∅ ↔ ∪ 𝐴 = ∅)
59	57, 58	bitri 278	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 = ∅)
60		pw0 4705	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
61	60	sseq2i 3944	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ {∅})
62		sssn 4719	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
63	61, 62	bitri 278	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
64		df-ne 2988	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
65		0ex 5175	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
66	65	unisn 4820	. . . . . . . . . . . . . . . 16 ⊢ ∪ {∅} = ∅
67	65	snid 4561	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅}
68	66, 67	eqeltri 2886	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} ∈ {∅}
69		unieq 4811	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = {∅} → ∪ 𝐴 = ∪ {∅})
70		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = {∅} → 𝐴 = {∅})
71	69, 70	eleq12d 2884	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = {∅} → (∪ 𝐴 ∈ 𝐴 ↔ ∪ {∅} ∈ {∅}))
72	68, 71	mpbiri 261	. . . . . . . . . . . . . 14 ⊢ (𝐴 = {∅} → ∪ 𝐴 ∈ 𝐴)
73	72	orim2i 908	. . . . . . . . . . . . 13 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 = ∅ ∨ ∪ 𝐴 ∈ 𝐴))
74	73	ord 861	. . . . . . . . . . . 12 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (¬ 𝐴 = ∅ → ∪ 𝐴 ∈ 𝐴))
75	64, 74	syl5bi 245	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴))
76	63, 75	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝒫 ∅ → (𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴))
77	59, 76	sylbir 238	. . . . . . . . 9 ⊢ (∪ 𝐴 = ∅ → (𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴))
78	77	com12 32	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (∪ 𝐴 = ∅ → ∪ 𝐴 ∈ 𝐴))
79	78	con3d 155	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → (¬ ∪ 𝐴 ∈ 𝐴 → ¬ ∪ 𝐴 = ∅))
80	56, 55, 79	sylc 65	. . . . . 6 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ ∪ 𝐴 = ∅)
81		ioran 981	. . . . . 6 ⊢ (¬ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅) ↔ (¬ ∪ 𝐴 ∈ 𝐴 ∧ ¬ ∪ 𝐴 = ∅))
82	55, 80, 81	sylanbrc 586	. . . . 5 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅))
83		uniun 4823	. . . . . . . 8 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
84	66	uneq2i 4087	. . . . . . . 8 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
85		un0 4298	. . . . . . . 8 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
86	83, 84, 85	3eqtri 2825	. . . . . . 7 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
87	86	eleq1i 2880	. . . . . 6 ⊢ (∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ ∪ 𝐴 ∈ (𝐴 ∪ {∅}))
88		elun 4076	. . . . . 6 ⊢ (∪ 𝐴 ∈ (𝐴 ∪ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ {∅}))
89	65	elsn2 4564	. . . . . . 7 ⊢ (∪ 𝐴 ∈ {∅} ↔ ∪ 𝐴 = ∅)
90	89	orbi2i 910	. . . . . 6 ⊢ ((∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅))
91	87, 88, 90	3bitri 300	. . . . 5 ⊢ (∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅))
92	82, 91	sylnibr 332	. . . 4 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ ∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}))
93		unieq 4811	. . . . . 6 ⊢ (ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = ∪ (𝐴 ∪ {∅}))
94		id 22	. . . . . 6 ⊢ (ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
95	93, 94	eleq12d 2884	. . . . 5 ⊢ (ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → (∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ↔ ∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
96	95	notbid 321	. . . 4 ⊢ (ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → (¬ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ↔ ¬ ∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})))
97	92, 96	syl5ibrcom 250	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → (ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → ¬ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})))
98	54, 97	mpd 15	. 2 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}))
99	50, 98	pm2.65da 816	1 ⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ Fin^III)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {crab 3110 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 Or wor 5437 ran crn 5520 suc csuc 6161 ⟶wf 6320 ‘cfv 6324 [⊊] crpss 7428 ωcom 7560 ≼ cdom 8490 Fincfn 8492 Fin^IIIcfin3 9692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-rpss 7429 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wdom 9013 df-card 9352 df-fin4 9698 df-fin3 9699

This theorem is referenced by: fin1a2s 9825

W3C validator