Theorem fin1a2lem11 9832

Description: Lemma for fin1a2 9837. (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 2821	. . 3 ⊢ (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
2	1	rnmpt 5827	. 2 ⊢ ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}}
3		unieq 4849	. . . . . . . . . . . 12 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ ∅)
4		uni0 4866	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
5	3, 4	syl6eq 2872	. . . . . . . . . . 11 ⊢ ({𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
6	5	adantl 484	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅)
7		0ex 5211	. . . . . . . . . . 11 ⊢ ∅ ∈ V
8	7	elsn2 4604	. . . . . . . . . 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 870	. . . . . . . 8 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∅) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
11		ssrab2 4056	. . . . . . . . . 10 ⊢ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ⊆ 𝐴
12		simpr 487	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅)
13		fin1a2lem9 9830	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin ∧ 𝑏 ∈ ω) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
14	13	ad4ant123 1168	. . . . . . . . . . 11 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ Fin)
15		simplll 773	. . . . . . . . . . . 12 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → [⊊] Or 𝐴)
16		soss 5493	. . . . . . . . . . . 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 9831	. . . . . . . . . . 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 1367	. . . . . . . . . 10 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
20	11, 19	sseldi 3965	. . . . . . . . 9 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) ∧ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ≠ ∅) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴)
21	20	orcd 869	. . . . . . . 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 3104	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑏 ∈ ω) → (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴 ∨ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
23		eleq1 2900	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ 𝐴))
24		eleq1 2900	. . . . . . . 8 ⊢ (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ {∅} ↔ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ∈ {∅}))
25	23, 24	orbi12d 915	. . . . . . 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 3284	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} → (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
28		simpr 487	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
29	28	sselda 3967	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ Fin)
30		ficardom 9390	. . . . . . . . 9 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ∈ ω)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ∈ ω)
32		breq1 5069	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑑 → (𝑐 ≼ (card‘𝑑) ↔ 𝑑 ≼ (card‘𝑑)))
33		simpr 487	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
34		ficardid 9391	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ Fin → (card‘𝑑) ≈ 𝑑)
35	29, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (card‘𝑑) ≈ 𝑑)
36		ensym 8558	. . . . . . . . . . . 12 ⊢ ((card‘𝑑) ≈ 𝑑 → 𝑑 ≈ (card‘𝑑))
37		endom 8536	. . . . . . . . . . . 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 3682	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
40		elssuni 4868	. . . . . . . . . 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 5069	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ (card‘𝑑) ↔ 𝑏 ≼ (card‘𝑑)))
43	42	elrab 3680	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)))
44		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ (card‘𝑑))
45	35	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (card‘𝑑) ≈ 𝑑)
46		domentr 8568	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ≼ (card‘𝑑) ∧ (card‘𝑑) ≈ 𝑑) → 𝑏 ≼ 𝑑)
47	44, 45, 46	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ≼ 𝑑)
48		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝐴 ⊆ Fin)
49		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ 𝐴)
50	48, 49	sseldd 3968	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑏 ∈ Fin)
51	29	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ Fin)
52		simplll 773	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → [⊊] Or 𝐴)
53		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → 𝑑 ∈ 𝐴)
54		sorpssi 7455	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑏 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
55	52, 49, 53, 54	syl12anc 834	. . . . . . . . . . . . . . 15 ⊢ (((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) ∧ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑))) → (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏))
56		fincssdom 9745	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ 𝑑 ∈ Fin ∧ (𝑏 ⊆ 𝑑 ∨ 𝑑 ⊆ 𝑏)) → (𝑏 ≼ 𝑑 ↔ 𝑏 ⊆ 𝑑))
57	50, 51, 55, 56	syl3anc 1367	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . 12 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ (card‘𝑑)) → 𝑏 ⊆ 𝑑))
60	43, 59	syl5bi 244	. . . . . . . . . . 11 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)} → 𝑏 ⊆ 𝑑))
61	60	ralrimiv 3181	. . . . . . . . . 10 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∀𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}𝑏 ⊆ 𝑑)
62		unissb 4870	. . . . . . . . . 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 3984	. . . . . . . 8 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
65		breq2 5070	. . . . . . . . . . 11 ⊢ (𝑏 = (card‘𝑑) → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ (card‘𝑑)))
66	65	rabbidv 3480	. . . . . . . . . 10 ⊢ (𝑏 = (card‘𝑑) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
67	66	unieqd 4852	. . . . . . . . 9 ⊢ (𝑏 = (card‘𝑑) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)})
68	67	rspceeqv 3638	. . . . . . . 8 ⊢ (((card‘𝑑) ∈ ω ∧ 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ (card‘𝑑)}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
69	31, 64, 68	syl2anc 586	. . . . . . 7 ⊢ ((( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) ∧ 𝑑 ∈ 𝐴) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
70	69	ex 415	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ 𝐴 → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
71		velsn 4583	. . . . . . 7 ⊢ (𝑑 ∈ {∅} ↔ 𝑑 = ∅)
72		peano1 7601	. . . . . . . . 9 ⊢ ∅ ∈ ω
73		dom0 8645	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ≼ ∅ ↔ 𝑏 = ∅)
74	73	biimpi 218	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ≼ ∅ → 𝑏 = ∅)
75	74	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅)
76	75	a1i 11	. . . . . . . . . . . . 13 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅) → 𝑏 = ∅))
77		breq1 5069	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (𝑐 ≼ ∅ ↔ 𝑏 ≼ ∅))
78	77	elrab 3680	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ↔ (𝑏 ∈ 𝐴 ∧ 𝑏 ≼ ∅))
79		velsn 4583	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {∅} ↔ 𝑏 = ∅)
80	76, 78, 79	3imtr4g 298	. . . . . . . . . . . 12 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑏 ∈ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} → 𝑏 ∈ {∅}))
81	80	ssrdv 3973	. . . . . . . . . . 11 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
82		uni0b 4864	. . . . . . . . . . 11 ⊢ (∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅ ↔ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} ⊆ {∅})
83	81, 82	sylibr 236	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅} = ∅)
84	83	eqcomd 2827	. . . . . . . . 9 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
85		breq2 5070	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (𝑐 ≼ 𝑏 ↔ 𝑐 ≼ ∅))
86	85	rabbidv 3480	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
87	86	unieqd 4852	. . . . . . . . . 10 ⊢ (𝑏 = ∅ → ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅})
88	87	rspceeqv 3638	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ ∅}) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
89	72, 84, 88	sylancr 589	. . . . . . . 8 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏})
90		eqeq1 2825	. . . . . . . . 9 ⊢ (𝑑 = ∅ → (𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
91	90	rexbidv 3297	. . . . . . . 8 ⊢ (𝑑 = ∅ → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ ∃𝑏 ∈ ω ∅ = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
92	89, 91	syl5ibrcom 249	. . . . . . 7 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 = ∅ → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
93	71, 92	syl5bi 244	. . . . . 6 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (𝑑 ∈ {∅} → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
94	70, 93	jaod 855	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ((𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}) → ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}))
95	27, 94	impbid 214	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅})))
96		elun 4125	. . . 4 ⊢ (𝑑 ∈ (𝐴 ∪ {∅}) ↔ (𝑑 ∈ 𝐴 ∨ 𝑑 ∈ {∅}))
97	95, 96	syl6bbr 291	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → (∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏} ↔ 𝑑 ∈ (𝐴 ∪ {∅})))
98	97	abbi1dv 2952	. 2 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → {𝑑 ∣ ∃𝑏 ∈ ω 𝑑 = ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}} = (𝐴 ∪ {∅}))
99	2, 98	syl5eq 2868	1 ⊢ (( [⊊] Or 𝐴 ∧ 𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ ∪ {𝑐 ∈ 𝐴 ∣ 𝑐 ≼ 𝑏}) = (𝐴 ∪ {∅}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  {csn 4567  ∪ cuni 4838   class class class wbr 5066   ↦ cmpt 5146   Or wor 5473  ran crn 5556  ‘cfv 6355   [⊊] crpss 7448  ωcom 7580   ≈ cen 8506   ≼ cdom 8507  Fincfn 8509  cardccrd 9364

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-rpss 7449  df-om 7581  df-1o 8102  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368

This theorem is referenced by:  fin1a2lem12  9833