Theorem fin1a2lem11 10332

Description: Lemma for fin1a2 10337. (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 2737	. . 3 ⊢ (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
2	1	rnmpt 5914	. 2 ⊢ ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}}
3		unieq 4876	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ ∅)
4		uni0 4893	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
5	3, 4	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
6	5	adantl 481	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
7		0ex 5254	. . . . . . . . . . 11 ⊢ ∅ ∈ V
8	7	elsn2 4624	. . . . . . . . . 10 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
9	6, 8	sylibr 234	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅})
10	9	olcd 875	. . . . . . . 8 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
11		ssrab2 4034	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ⊆ 𝐴
12		simpr 484	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅)
13		fin1a2lem9 10330	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
14	13	ad4ant123 1174	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
15		simplll 775	. . . . . . . . . . . 12 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → [⊊] Or 𝐴)
16		soss 5560	. . . . . . . . . . . 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 10331	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
20	11, 19	sselid 3933	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴)
21	20	orcd 874	. . . . . . . 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 3020	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
23		eleq1 2825	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴))
24		eleq1 2825	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
25	23, 24	orbi12d 919	. . . . . . 7 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) ↔ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅})))
26	22, 25	syl5ibrcom 247	. . . . . 6 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
27	26	rexlimdva 3139	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
28		simpr 484	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
29	28	sselda 3935	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ Fin)
30		ficardom 9885	. . . . . . . . 9 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ∈ ω)
32		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
33		simpr 484	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
34		ficardid 9886	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
35	29, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ≈ 𝑑)
36		ensym 8952	. . . . . . . . . . . 12 ⊢ ((card‘𝑑) ≈ 𝑑 → 𝑑 ≈ (card‘𝑑))
37		endom 8928	. . . . . . . . . . . 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 3650	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
40		elssuni 4896	. . . . . . . . . 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 5103	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
45	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
46		domentr 8962	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏 ≼ 𝑑)
47	44, 45, 46	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ 𝑑)
48		simpllr 776	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
49		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ 𝐴)
50	48, 49	sseldd 3936	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
51	29	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
52		simplll 775	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → [⊊] Or 𝐴)
53		simplr 769	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ 𝐴)
54		sorpssi 7684	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑏 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
55	52, 49, 53, 54	syl12anc 837	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
56		fincssdom 10245	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏)) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
57	50, 51, 55, 56	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
58	47, 57	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ⊆ 𝑑)
59	58	ex 412	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)) → 𝑏 ⊆ 𝑑))
60	43, 59	biimtrid 242	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} → 𝑏 ⊆ 𝑑))
61	60	ralrimiv 3129	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
62		unissb 4898	. . . . . . . . . 10 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ⊆ 𝑑 ↔ ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
63	61, 62	sylibr 234	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ⊆ 𝑑)
64	41, 63	eqssd 3953	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
65		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑏 = (card‘𝑑) → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ (card‘𝑑)))
66	65	rabbidv 3408	. . . . . . . . . 10 ⊢ (𝑏 = (card‘𝑑) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
67	66	unieqd 4878	. . . . . . . . 9 ⊢ (𝑏 = (card‘𝑑) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
68	67	rspceeqv 3601	. . . . . . . 8 ⊢ (((card‘𝑑) ∈ ω ∧ 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
69	31, 64, 68	syl2anc 585	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
70	69	ex 412	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ 𝐴 → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
71		velsn 4598	. . . . . . 7 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
72		peano1 7841	. . . . . . . . 9 ⊢ ∅ ∈ ω
73		dom0 9045	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
74	73	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ≼ ∅ → 𝑏 = ∅)
75	74	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅)
76	75	a1i 11	. . . . . . . . . . . . 13 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅))
77		breq1 5103	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
78	77	elrab 3648	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅))
79		velsn 4598	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
80	76, 78, 79	3imtr4g 296	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
81	80	ssrdv 3941	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
82		uni0b 4891	. . . . . . . . . . 11 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅ ↔ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
83	81, 82	sylibr 234	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅)
84	83	eqcomd 2743	. . . . . . . . 9 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
85		breq2 5104	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ ∅))
86	85	rabbidv 3408	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
87	86	unieqd 4878	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
88	87	rspceeqv 3601	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
89	72, 84, 88	sylancr 588	. . . . . . . 8 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
90		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑑 = ∅ → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
91	90	rexbidv 3162	. . . . . . . 8 ⊢ (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
92	89, 91	syl5ibrcom 247	. . . . . . 7 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
93	71, 92	biimtrid 242	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
94	70, 93	jaod 860	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
95	27, 94	impbid 212	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
96		elun 4107	. . . 4 ⊢ (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}))
97	95, 96	bitr4di 289	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
98	97	eqabcdv 2871	. 2 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}} = (𝐴 ∪ {∅}))
99	2, 98	eqtrid 2784	1 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181   Or wor 5539  ran crn 5633  ‘cfv 6500   [⊊] crpss 7677  ωcom 7818   ≈ cen 8892   ≼ cdom 8893  Fincfn 8895  cardccrd 9859

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-rpss 7678  df-om 7819  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863

This theorem is referenced by:  fin1a2lem12  10333