Theorem zornn0g 10529

Description: Variant of Zorn's lemma zorng 10528 in which ∅, the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
zornn0g	⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem zornn0g

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1135	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ≠ ∅)
2		simp1 1134	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ∈ dom card)
3		snfi 9069	. . . . 5 ⊢ {∅} ∈ Fin
4		finnum 9972	. . . . 5 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
5	3, 4	ax-mp 5	. . . 4 ⊢ {∅} ∈ dom card
6		unnum 10220	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
7	2, 5, 6	sylancl 585	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8		uncom 4152	. . . . . . . . 9 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
9	8	sseq2i 4009	. . . . . . . 8 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10		ssundif 4488	. . . . . . . 8 ⊢ (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
11	9, 10	bitri 275	. . . . . . 7 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12		difss 4130	. . . . . . . . 9 ⊢ (𝑤 ∖ {∅}) ⊆ 𝑤
13		soss 5610	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅})))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅}))
15		ssdif0 4364	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16		uni0b 4936	. . . . . . . . . . . . 13 ⊢ (∪ 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
17	16	biimpri 227	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ {∅} → ∪ 𝑤 = ∅)
18	17	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
19	15, 18	sylbir 234	. . . . . . . . . 10 ⊢ ((𝑤 ∖ {∅}) = ∅ → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
20	19	imbi2d 340	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21		vex 3475	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
22	21	difexi 5330	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {∅}) ∈ V
23		sseq1 4005	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ⊆ 𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24		neeq1 3000	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25		soeq2 5612	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ( [⊊] Or 𝑧 ↔ [⊊] Or (𝑤 ∖ {∅})))
26	23, 24, 25	3anbi123d 1433	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅}))))
27		unieq 4919	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ∪ 𝑧 = ∪ (𝑤 ∖ {∅}))
28	27	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (∪ 𝑧 ∈ 𝐴 ↔ ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
29	26, 28	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) ↔ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴)))
30	22, 29	spcv 3592	. . . . . . . . . . . . 13 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
31	30	com12 32	. . . . . . . . . . . 12 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
32	31	3expa 1116	. . . . . . . . . . 11 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
33	32	an32s 651	. . . . . . . . . 10 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
34		unidif0 5360	. . . . . . . . . . . 12 ⊢ ∪ (𝑤 ∖ {∅}) = ∪ 𝑤
35	34	eleq1i 2820	. . . . . . . . . . 11 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 ↔ ∪ 𝑤 ∈ 𝐴)
36		elun1 4176	. . . . . . . . . . 11 ⊢ (∪ 𝑤 ∈ 𝐴 → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))
37	35, 36	sylbi 216	. . . . . . . . . 10 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))
38	33, 37	syl6 35	. . . . . . . . 9 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
39		0ex 5307	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
40	39	snid 4665	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
41		elun2 4177	. . . . . . . . . . 11 ⊢ (∅ ∈ {∅} → ∅ ∈ (𝐴 ∪ {∅}))
42	40, 41	ax-mp 5	. . . . . . . . . 10 ⊢ ∅ ∈ (𝐴 ∪ {∅})
43	42	2a1i 12	. . . . . . . . 9 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∅ ∈ (𝐴 ∪ {∅})))
44	20, 38, 43	pm2.61ne 3024	. . . . . . . 8 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
45	14, 44	sylan2 592	. . . . . . 7 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
46	11, 45	sylanb 580	. . . . . 6 ⊢ ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
47	46	com12 32	. . . . 5 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
48	47	alrimiv 1923	. . . 4 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
49	48	3ad2ant3 1133	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
50		zorng 10528	. . 3 ⊢ (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
51	7, 49, 50	syl2anc 583	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
52		ssun1 4172	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ {∅})
53		ssralv 4048	. . . . 5 ⊢ (𝐴 ⊆ (𝐴 ∪ {∅}) → (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
54	52, 53	ax-mp 5	. . . 4 ⊢ (∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
55	54	reximi 3081	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
56		rexun 4190	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
57		simpr 484	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
58		simpr 484	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
59		psseq1 4085	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ ∅ ⊊ 𝑦))
60		0pss 4445	. . . . . . . . . . . . 13 ⊢ (∅ ⊊ 𝑦 ↔ 𝑦 ≠ ∅)
61	59, 60	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ 𝑦 ≠ ∅))
62	61	notbid 318	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ 𝑦 ≠ ∅))
63		nne 2941	. . . . . . . . . . 11 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
64	62, 63	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ 𝑦 = ∅))
65	64	ralbidv 3174	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
66	39, 65	rexsn 4687	. . . . . . . 8 ⊢ (∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅)
67		eqsn 4833	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
68	67	biimpar 477	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑦 = ∅) → 𝐴 = {∅})
69	66, 68	sylan2b 593	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → 𝐴 = {∅})
70	69	rexeqdv 3323	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
71	58, 70	mpbird 257	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
72	57, 71	jaodan 956	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
73	56, 72	sylan2b 593	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
74	55, 73	sylan2 592	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
75	1, 51, 74	syl2anc 583	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085  ∀wal 1532   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  ∀wral 3058  ∃wrex 3067   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947   ⊊ wpss 3948  ∅c0 4323  {csn 4629  ∪ cuni 4908   Or wor 5589  dom cdm 5678   [⊊] crpss 7727  Fincfn 8964  cardccrd 9959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7728  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-en 8965  df-dom 8966  df-fin 8968  df-dju 9925  df-card 9963

This theorem is referenced by:  zornn0  10532  pgpfac1lem5  20036  lbsextlem4  21049  filssufilg  23828