Theorem fin1a2lem11 10361

Description: Lemma for fin1a2 10366. (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 2761	. . 3 ⊢ (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
2	1	rnmpt 5929	. 2 ⊢ ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}}
3		unieq 4873	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ ∅)
4		uni0 4891	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
5	3, 4	eqtrdi 2812	. . . . . . . . . . 11 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
6	5	adantl 485	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
7		0ex 5254	. . . . . . . . . . 11 ⊢ ∅ ∈ V
8	7	elsn2 4621	. . . . . . . . . 10 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
9	6, 8	sylibr 236	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅})
10	9	olcd 885	. . . . . . . 8 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
11		ssrab2 4031	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ⊆ 𝐴
12		simpr 488	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅)
13		fin1a2lem9 10359	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
14	13	ad4ant123 1185	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
15		simplll 784	. . . . . . . . . . . 12 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → [⊊] Or 𝐴)
16		soss 5571	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ⊆ 𝐴 → ( [⊊] Or 𝐴 → [⊊] Or {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
17	11, 15, 16	mpsyl 68	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → [⊊] Or {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
18		fin1a2lem10 10360	. . . . . . . . . . 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 1389	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
20	11, 19	sselid 3932	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴)
21	20	orcd 884	. . . . . . . 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 3043	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
23		eleq1 2849	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴))
24		eleq1 2849	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
25	23, 24	orbi12d 929	. . . . . . 7 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) ↔ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅})))
26	22, 25	syl5ibrcom 249	. . . . . 6 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
27	26	rexlimdva 3162	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
28		simpr 488	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
29	28	sselda 3934	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ Fin)
30		ficardom 9913	. . . . . . . . 9 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ∈ ω)
32		breq1 5100	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
33		simpr 488	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
34		ficardid 9914	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
35	29, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ≈ 𝑑)
36		ensym 8978	. . . . . . . . . . . 12 ⊢ ((card‘𝑑) ≈ 𝑑 → 𝑑 ≈ (card‘𝑑))
37		endom 8954	. . . . . . . . . . . 12 ⊢ (𝑑 ≈ (card‘𝑑) → 𝑑 ≼ (card‘𝑑))
38	35, 36, 37	3syl 18	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ≼ (card‘𝑑))
39	32, 33, 38	elrabd 3651	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
40		elssuni 4894	. . . . . . . . . 10 ⊢ (𝑑 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} → 𝑑 ⊆ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
41	39, 40	syl 17	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ⊆ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
42		breq1 5100	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
43	42	elrab 3649	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)))
44		simprr 782	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
45	35	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
46		domentr 8988	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏 ≼ 𝑑)
47	44, 45, 46	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ 𝑑)
48		simpllr 785	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
49		simprl 780	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ 𝐴)
50	48, 49	sseldd 3935	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
51	29	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
52		simplll 784	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → [⊊] Or 𝐴)
53		simplr 778	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ 𝐴)
54		sorpssi 7707	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑏 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
55	52, 49, 53, 54	syl12anc 847	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
56		fincssdom 10274	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏)) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
57	50, 51, 55, 56	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
58	47, 57	mpbid 234	. . . . . . . . . . . . 13 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ⊆ 𝑑)
59	58	ex 416	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)) → 𝑏 ⊆ 𝑑))
60	43, 59	biimtrid 244	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} → 𝑏 ⊆ 𝑑))
61	60	ralrimiv 3152	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
62		unissb 4896	. . . . . . . . . 10 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
63	61, 62	sylibr 236	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
64	41, 63	eqssd 3951	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
65		breq2 5101	. . . . . . . . . . 11 ⊢ (𝑏 = (card‘𝑑) → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ (card‘𝑑)))
66	65	rabbidv 3420	. . . . . . . . . 10 ⊢ (𝑏 = (card‘𝑑) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
67	66	unieqd 4875	. . . . . . . . 9 ⊢ (𝑏 = (card‘𝑑) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
68	67	rspceeqv 3603	. . . . . . . 8 ⊢ (((card‘𝑑) ∈ ω ∧ 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
69	31, 64, 68	syl2anc 593	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
70	69	ex 416	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ 𝐴 → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
71		velsn 4595	. . . . . . 7 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
72		peano1 7864	. . . . . . . . 9 ⊢ ∅ ∈ ω
73		dom0 9071	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
74	73	bilani 508	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅)
75	74	a1i 11	. . . . . . . . . . . . 13 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅))
76		breq1 5100	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
77	76	elrab 3649	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅))
78		velsn 4595	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
79	75, 77, 78	3imtr4g 298	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
80	79	ssrdv 3940	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
81		uni0b 4889	. . . . . . . . . . 11 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅ ↔ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
82	80, 81	sylibr 236	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅)
83	82	eqcomd 2767	. . . . . . . . 9 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
84		breq2 5101	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ ∅))
85	84	rabbidv 3420	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
86	85	unieqd 4875	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
87	86	rspceeqv 3603	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
88	72, 83, 87	sylancr 596	. . . . . . . 8 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
89		eqeq1 2765	. . . . . . . . 9 ⊢ (𝑑 = ∅ → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
90	89	rexbidv 3185	. . . . . . . 8 ⊢ (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
91	88, 90	syl5ibrcom 249	. . . . . . 7 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
92	71, 91	biimtrid 244	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
93	70, 92	jaod 870	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
94	27, 93	impbid 214	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
95		elun 4104	. . . 4 ⊢ (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}))
96	94, 95	bitr4di 291	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
97	96	eqabcdv 2895	. 2 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}} = (𝐴 ∪ {∅}))
98	2, 97	eqtrid 2808	1 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  {crab 3413   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  {csn 4579  ∪ cuni 4862   class class class wbr 5097   ↦ cmpt 5178   Or wor 5550  ran crn 5644  ‘cfv 6516   [⊊] crpss 7700  ωcom 7841   ≈ cen 8918   ≼ cdom 8919  Fincfn 8921  cardccrd 9887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-rpss 7701  df-om 7842  df-1o 8431  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9891

This theorem is referenced by:  fin1a2lem12  10362