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Theorem fin1a2lem12 10449

Description: Lemma for fin1a2 10453. (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 484	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐵 ∈ Fin^III)
2		simpll1 1211	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ⊆ 𝒫 𝐵)
3	2	adantr 480	. . . . . 6 ⊢ (((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) ∧ 𝑒 ∈ ω) → 𝐴 ⊆ 𝒫 𝐵)
4		ssrab2 4090	. . . . . . . 8 ⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐴
5	4	unissi 4921	. . . . . . 7 ⊢ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ ∪ 𝐴
6		sspwuni 5105	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
7	6	biimpi 216	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)
8	5, 7	sstrid 4007	. . . . . 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 5339	. . . . . 6 ⊢ (𝐵 ∈ Fin^III → (∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵 ↔ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵))
11	10	ad2antlr 727	. . . . 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 257	. . . 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 7135	. . 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 3482	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
15	14	sucex 7826	. . . . . . . . . 10 ⊢ suc 𝑑 ∈ V
16		sssucid 6466	. . . . . . . . . 10 ⊢ 𝑑 ⊆ suc 𝑑
17		ssdomg 9039	. . . . . . . . . 10 ⊢ (suc 𝑑 ∈ V → (𝑑 ⊆ suc 𝑑 → 𝑑 ≼ suc 𝑑))
18	15, 16, 17	mp2 9	. . . . . . . . 9 ⊢ 𝑑 ≼ suc 𝑑
19		domtr 9046	. . . . . . . . 9 ⊢ ((𝑓 ≼ 𝑑 ∧ 𝑑 ≼ suc 𝑑) → 𝑓 ≼ suc 𝑑)
20	18, 19	mpan2 691	. . . . . . . 8 ⊢ (𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑)
21	20	a1i 11	. . . . . . 7 ⊢ (𝑓 ∈ 𝐴 → (𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑))
22	21	ss2rabi 4087	. . . . . 6 ⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}
23		uniss 4920	. . . . . 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 5384	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin^III → 𝒫 𝐵 ∈ V)
27	26	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝒫 𝐵 ∈ V)
28	27, 2	ssexd 5330	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ∈ V)
29		rabexg 5343	. . . . . . 7 ⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V)
30		uniexg 7759	. . . . . . 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 5152	. . . . . . . . 9 ⊢ (𝑒 = 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ 𝑑))
33	32	rabbidv 3441	. . . . . . . 8 ⊢ (𝑒 = 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
34	33	unieqd 4925	. . . . . . 7 ⊢ (𝑒 = 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
35		eqid 2735	. . . . . . 7 ⊢ (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})
36	34, 35	fvmptg 7014	. . . . . 6 ⊢ ((𝑑 ∈ ω ∧ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
37	25, 31, 36	syl2anr 597	. . . . 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 7913	. . . . . 6 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
39		rabexg 5343	. . . . . . 7 ⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V)
40		uniexg 7759	. . . . . . 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 5152	. . . . . . . . 9 ⊢ (𝑒 = suc 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ suc 𝑑))
43	42	rabbidv 3441	. . . . . . . 8 ⊢ (𝑒 = suc 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
44	43	unieqd 4925	. . . . . . 7 ⊢ (𝑒 = suc 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
45	44, 35	fvmptg 7014	. . . . . 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 597	. . . . 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 4043	. . . 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 3144	. . 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 10419	. . 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 1370	. 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 10448	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
52	51	adantrr 717	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
53	52	3ad2antl2 1185	. . . 4 ⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
54	53	adantr 480	. . 3 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ran (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}))
55		simpll3 1213	. . . . . 6 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ ∪ 𝐴 ∈ 𝐴)
56		simplrr 778	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ≠ ∅)
57		sspwuni 5105	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 ⊆ ∅)
58		ss0b 4407	. . . . . . . . . . 11 ⊢ (∪ 𝐴 ⊆ ∅ ↔ ∪ 𝐴 = ∅)
59	57, 58	bitri 275	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 = ∅)
60		pw0 4817	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
61	60	sseq2i 4025	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ {∅})
62		sssn 4831	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
63	61, 62	bitri 275	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
64		df-ne 2939	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
65		0ex 5313	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
66	65	unisn 4931	. . . . . . . . . . . . . . . 16 ⊢ ∪ {∅} = ∅
67	65	snid 4667	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅}
68	66, 67	eqeltri 2835	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} ∈ {∅}
69		unieq 4923	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = {∅} → ∪ 𝐴 = ∪ {∅})
70		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = {∅} → 𝐴 = {∅})
71	69, 70	eleq12d 2833	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = {∅} → (∪ 𝐴 ∈ 𝐴 ↔ ∪ {∅} ∈ {∅}))
72	68, 71	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝐴 = {∅} → ∪ 𝐴 ∈ 𝐴)
73	72	orim2i 910	. . . . . . . . . . . . 13 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 = ∅ ∨ ∪ 𝐴 ∈ 𝐴))
74	73	ord 864	. . . . . . . . . . . 12 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (¬ 𝐴 = ∅ → ∪ 𝐴 ∈ 𝐴))
75	64, 74	biimtrid 242	. . . . . . . . . . 11 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴))
76	63, 75	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝒫 ∅ → (𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴))
77	59, 76	sylbir 235	. . . . . . . . 9 ⊢ (∪ 𝐴 = ∅ → (𝐴 ≠ ∅ → ∪ 𝐴 ∈ 𝐴))
78	77	com12 32	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (∪ 𝐴 = ∅ → ∪ 𝐴 ∈ 𝐴))
79	78	con3d 152	. . . . . . 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 985	. . . . . 6 ⊢ (¬ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅) ↔ (¬ ∪ 𝐴 ∈ 𝐴 ∧ ¬ ∪ 𝐴 = ∅))
82	55, 80, 81	sylanbrc 583	. . . . 5 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅))
83		uniun 4935	. . . . . . . 8 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
84	66	uneq2i 4175	. . . . . . . 8 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
85		un0 4400	. . . . . . . 8 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
86	83, 84, 85	3eqtri 2767	. . . . . . 7 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
87	86	eleq1i 2830	. . . . . 6 ⊢ (∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ ∪ 𝐴 ∈ (𝐴 ∪ {∅}))
88		elun 4163	. . . . . 6 ⊢ (∪ 𝐴 ∈ (𝐴 ∪ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ {∅}))
89	65	elsn2 4670	. . . . . . 7 ⊢ (∪ 𝐴 ∈ {∅} ↔ ∪ 𝐴 = ∅)
90	89	orbi2i 912	. . . . . 6 ⊢ ((∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅))
91	87, 88, 90	3bitri 297	. . . . 5 ⊢ (∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅))
92	82, 91	sylnibr 329	. . . 4 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → ¬ ∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}))
93		unieq 4923	. . . . . 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 2833	. . . . 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 318	. . . 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 247	. . 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 817	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 Or wor 5596 ran crn 5690 suc csuc 6388 ⟶wf 6559 ‘cfv 6563 [⊊] crpss 7741 ωcom 7887 ≼ cdom 8982 Fincfn 8984 Fin^IIIcfin3 10319

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-rpss 7742 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-wdom 9603 df-card 9977 df-fin4 10325 df-fin3 10326

This theorem is referenced by: fin1a2s 10452

W3C validator