Theorem fin1a2lem11 10527

Description: Lemma for fin1a2 10532. (Contributed by Stefan O'Rear, 8-Nov-2014.)

Assertion

Ref	Expression
fin1a2lem11	⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))

Distinct variable group:   𝑏,𝑐,𝐴

Proof of Theorem fin1a2lem11

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
2	1	rnmpt 5951	. 2 ⊢ ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}}
3		unieq 4883	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ ∅)
4		uni0 4901	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
5	3, 4	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
6	5	adantl 487	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
7		0ex 5267	. . . . . . . . . . 11 ⊢ ∅ ∈ V
8	7	elsn2 4627	. . . . . . . . . 10 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
9	6, 8	sylibr 237	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅})
10	9	olcd 888	. . . . . . . 8 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
11		ssrab2 4008	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ⊆ 𝐴
12		simpr 490	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅)
13		fin1a2lem9 10525	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
14	13	ad4ant123 1191	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
15		simplll 787	. . . . . . . . . . . 12 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → [⊊] Or 𝐴)
16		soss 5586	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ⊆ 𝐴 → ( [⊊] Or 𝐴 → [⊊] Or {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
17	11, 15, 16	mpsyl 69	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → [⊊] Or {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
18		fin1a2lem10 10526	. . . . . . . . . . 11 ⊢ (({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅ ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin ∧ [⊊] Or {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
19	12, 14, 17, 18	syl3anc 1399	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
20	11, 19	sselid 3911	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴)
21	20	orcd 887	. . . . . . . 8 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
22	10, 21	pm2.61dane 3046	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
23		eleq1 2852	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴))
24		eleq1 2852	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
25	23, 24	orbi12d 932	. . . . . . 7 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) ↔ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅})))
26	22, 25	syl5ibrcom 250	. . . . . 6 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
27	26	rexlimdva 3167	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
28		simpr 490	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
29	28	sselda 3913	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ Fin)
30		ficardom 10079	. . . . . . . . 9 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
31	29, 30	syl 18	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ∈ ω)
32		breq1 5111	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
33		simpr 490	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
34		ficardid 10080	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
35	29, 34	syl 18	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ≈ 𝑑)
36		ensym 9068	. . . . . . . . . . . 12 ⊢ ((card‘𝑑) ≈ 𝑑 → 𝑑 ≈ (card‘𝑑))
37		endom 9044	. . . . . . . . . . . 12 ⊢ (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
38	35, 36, 37	3syl 19	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ≼ (card‘𝑑))
39	32, 33, 38	elrabd 3650	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
40		elssuni 4904	. . . . . . . . . 10 ⊢ (𝑑 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} → 𝑑 ⊆ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
41	39, 40	syl 18	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ⊆ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
42		breq1 5111	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
43	42	elrab 3648	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)))
44		simprr 785	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
45	35	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
46		domentr 9078	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏 ≼ 𝑑)
47	44, 45, 46	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ 𝑑)
48		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
49		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ 𝐴)
50	48, 49	sseldd 3914	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
51	29	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
52		simplll 787	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → [⊊] Or 𝐴)
53		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ 𝐴)
54		sorpssi 7777	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑏 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
55	52, 49, 53, 54	syl12anc 850	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
56		fincssdom 10440	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏)) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
57	50, 51, 55, 56	syl3anc 1399	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
58	47, 57	mpbid 235	. . . . . . . . . . . . 13 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ⊆ 𝑑)
59	58	ex 418	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)) → 𝑏 ⊆ 𝑑))
60	43, 59	biimtrid 245	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} → 𝑏 ⊆ 𝑑))
61	60	ralrimiv 3157	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
62		unissb 4906	. . . . . . . . . 10 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
63	61, 62	sylibr 237	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
64	41, 63	eqssd 3930	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
65		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑏 = (card‘𝑑) → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ (card‘𝑑)))
66	65	rabbidv 3423	. . . . . . . . . 10 ⊢ (𝑏 = (card‘𝑑) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
67	66	unieqd 4885	. . . . . . . . 9 ⊢ (𝑏 = (card‘𝑑) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
68	67	rspceeqv 3602	. . . . . . . 8 ⊢ (((card‘𝑑) ∈ ω ∧ 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
69	31, 64, 68	syl2anc 596	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
70	69	ex 418	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ 𝐴 → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
71		velsn 4600	. . . . . . 7 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
72		peano1 7934	. . . . . . . . 9 ⊢ ∅ ∈ ω
73		dom0 9162	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
74	73	bilani 510	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅))
76		breq1 5111	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
77	76	elrab 3648	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅))
78		velsn 4600	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
79	75, 77, 78	3imtr4g 299	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
80	79	ssrdv 3919	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
81		uni0b 4899	. . . . . . . . . . 11 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅ ↔ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
82	80, 81	sylibr 237	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅)
83	82	eqcomd 2769	. . . . . . . . 9 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
84		breq2 5112	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ ∅))
85	84	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
86	85	unieqd 4885	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
87	86	rspceeqv 3602	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
88	72, 83, 87	sylancr 599	. . . . . . . 8 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
89		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑑 = ∅ → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
90	89	rexbidv 3190	. . . . . . . 8 ⊢ (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
91	88, 90	syl5ibrcom 250	. . . . . . 7 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
92	71, 91	biimtrid 245	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
93	70, 92	jaod 873	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
94	27, 93	impbid 215	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
95		elun 4155	. . . 4 ⊢ (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}))
96	94, 95	bitr4di 292	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
97	96	eqabcdv 2898	. 2 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}} = (𝐴 ∪ {∅}))
98	2, 97	eqtrid 2810	1 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ cuni 4872   class class class wbr 5108   ↦ cmpt 5192   Or wor 5565  ran crn 5659  ‘cfv 6566   [⊊] crpss 7770  ωcom 7911   ≈ cen 9008   ≼ cdom 9009  Fincfn 9011  cardccrd 10053

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-rpss 7771  df-om 7912  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057

This theorem is used by:  fin1a2lem12  10528