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Theorem fin1a2lem12 10311

Description: Lemma for fin1a2 10315. (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 1213	. . . . . . 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 4029	. . . . . . . 8 ⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐴
5	4	unissi 4869	. . . . . . 7 ⊢ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ ∪ 𝐴
6		sspwuni 5052	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
7	6	biimpi 216	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)
8	5, 7	sstrid 3942	. . . . . 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 5275	. . . . . 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 7056	. . 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 3441	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
15	14	sucex 7747	. . . . . . . . . 10 ⊢ suc 𝑑 ∈ V
16		sssucid 6395	. . . . . . . . . 10 ⊢ 𝑑 ⊆ suc 𝑑
17		ssdomg 8931	. . . . . . . . . 10 ⊢ (suc 𝑑 ∈ V → (𝑑 ⊆ suc 𝑑 → 𝑑 ≼ suc 𝑑))
18	15, 16, 17	mp2 9	. . . . . . . . 9 ⊢ 𝑑 ≼ suc 𝑑
19		domtr 8938	. . . . . . . . 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 4025	. . . . . 6 ⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}
23		uniss 4868	. . . . . 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 5320	. . . . . . . . 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 5266	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ Fin^III) → 𝐴 ∈ V)
29		rabexg 5279	. . . . . . 7 ⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V)
30		uniexg 7681	. . . . . . 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 5099	. . . . . . . . 9 ⊢ (𝑒 = 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ 𝑑))
33	32	rabbidv 3403	. . . . . . . 8 ⊢ (𝑒 = 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
34	33	unieqd 4873	. . . . . . 7 ⊢ (𝑒 = 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑})
35		eqid 2733	. . . . . . 7 ⊢ (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝑒 ∈ ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})
36	34, 35	fvmptg 6935	. . . . . 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 7828	. . . . . 6 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
39		rabexg 5279	. . . . . . 7 ⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V)
40		uniexg 7681	. . . . . . 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 5099	. . . . . . . . 9 ⊢ (𝑒 = suc 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ suc 𝑑))
43	42	rabbidv 3403	. . . . . . . 8 ⊢ (𝑒 = suc 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
44	43	unieqd 4873	. . . . . . 7 ⊢ (𝑒 = suc 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑})
45	44, 35	fvmptg 6935	. . . . . 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 3986	. . . 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 3125	. . 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 10281	. . 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 1373	. 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 10310	. . . . . 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 1187	. . . 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 1215	. . . . . 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 5052	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 ⊆ ∅)
58		ss0b 4350	. . . . . . . . . . 11 ⊢ (∪ 𝐴 ⊆ ∅ ↔ ∪ 𝐴 = ∅)
59	57, 58	bitri 275	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ ∪ 𝐴 = ∅)
60		pw0 4765	. . . . . . . . . . . . 13 ⊢ 𝒫 ∅ = {∅}
61	60	sseq2i 3960	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ 𝐴 ⊆ {∅})
62		sssn 4779	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
63	61, 62	bitri 275	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 ∅ ↔ (𝐴 = ∅ ∨ 𝐴 = {∅}))
64		df-ne 2930	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
65		0ex 5249	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
66	65	unisn 4879	. . . . . . . . . . . . . . . 16 ⊢ ∪ {∅} = ∅
67	65	snid 4616	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ {∅}
68	66, 67	eqeltri 2829	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} ∈ {∅}
69		unieq 4871	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = {∅} → ∪ 𝐴 = ∪ {∅})
70		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = {∅} → 𝐴 = {∅})
71	69, 70	eleq12d 2827	. . . . . . . . . . . . . . 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 4883	. . . . . . . 8 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
84	66	uneq2i 4114	. . . . . . . 8 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
85		un0 4343	. . . . . . . 8 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
86	83, 84, 85	3eqtri 2760	. . . . . . 7 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
87	86	eleq1i 2824	. . . . . 6 ⊢ (∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅}) ↔ ∪ 𝐴 ∈ (𝐴 ∪ {∅}))
88		elun 4102	. . . . . 6 ⊢ (∪ 𝐴 ∈ (𝐴 ∪ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈ {∅}))
89	65	elsn2 4619	. . . . . . 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 4871	. . . . . 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 2827	. . . . 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 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 {crab 3396 Vcvv 3437 ∪ cun 3896 ⊆ wss 3898 ∅c0 4282 𝒫 cpw 4551 {csn 4577 ∪ cuni 4860 class class class wbr 5095 ↦ cmpt 5176 Or wor 5528 ran crn 5622 suc csuc 6315 ⟶wf 6484 ‘cfv 6488 [⊊] crpss 7663 ωcom 7804 ≼ cdom 8875 Fincfn 8877 Fin^IIIcfin3 10181

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-rpss 7664 df-om 7805 df-2nd 7930 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-er 8630 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-wdom 9460 df-card 9841 df-fin4 10187 df-fin3 10188

This theorem is referenced by: fin1a2s 10314

W3C validator