Theorem zornn0g 10502

Description: Variant of Zorn's lemma zorng 10501 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 9046	. . . . 5 ⊢ {∅} ∈ Fin
4		finnum 9945	. . . . 5 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
5	3, 4	ax-mp 5	. . . 4 ⊢ {∅} ∈ dom card
6		unnum 10193	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
7	2, 5, 6	sylancl 584	. . 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 4010	. . . . . . . 8 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10		ssundif 4486	. . . . . . . 8 ⊢ (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
11	9, 10	bitri 274	. . . . . . 7 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12		difss 4130	. . . . . . . . 9 ⊢ (𝑤 ∖ {∅}) ⊆ 𝑤
13		soss 5607	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅})))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅}))
15		ssdif0 4362	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16		uni0b 4936	. . . . . . . . . . . . 13 ⊢ (∪ 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
17	16	biimpri 227	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ {∅} → ∪ 𝑤 = ∅)
18	17	eleq1d 2816	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
19	15, 18	sylbir 234	. . . . . . . . . 10 ⊢ ((𝑤 ∖ {∅}) = ∅ → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
20	19	imbi2d 339	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21		vex 3476	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
22	21	difexi 5327	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {∅}) ∈ V
23		sseq1 4006	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ⊆ 𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24		neeq1 3001	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25		soeq2 5609	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ( [⊊] Or 𝑧 ↔ [⊊] Or (𝑤 ∖ {∅})))
26	23, 24, 25	3anbi123d 1434	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅}))))
27		unieq 4918	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ∪ 𝑧 = ∪ (𝑤 ∖ {∅}))
28	27	eleq1d 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (∪ 𝑧 ∈ 𝐴 ↔ ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
29	26, 28	imbi12d 343	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) ↔ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴)))
30	22, 29	spcv 3594	. . . . . . . . . . . . 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 648	. . . . . . . . . 10 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
34		unidif0 5357	. . . . . . . . . . . 12 ⊢ ∪ (𝑤 ∖ {∅}) = ∪ 𝑤
35	34	eleq1i 2822	. . . . . . . . . . 11 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 ↔ ∪ 𝑤 ∈ 𝐴)
36		elun1 4175	. . . . . . . . . . 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 5306	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
40	39	snid 4663	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
41		elun2 4176	. . . . . . . . . . 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 3025	. . . . . . . 8 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
45	14, 44	sylan2 591	. . . . . . 7 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
46	11, 45	sylanb 579	. . . . . 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 1928	. . . 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 10501	. . 3 ⊢ (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
51	7, 49, 50	syl2anc 582	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
52		ssun1 4171	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ {∅})
53		ssralv 4049	. . . . 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 3082	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
56		rexun 4189	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
57		simpr 483	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
58		simpr 483	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
59		psseq1 4086	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ ∅ ⊊ 𝑦))
60		0pss 4443	. . . . . . . . . . . . 13 ⊢ (∅ ⊊ 𝑦 ↔ 𝑦 ≠ ∅)
61	59, 60	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ 𝑦 ≠ ∅))
62	61	notbid 317	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ 𝑦 ≠ ∅))
63		nne 2942	. . . . . . . . . . 11 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
64	62, 63	bitrdi 286	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ 𝑦 = ∅))
65	64	ralbidv 3175	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
66	39, 65	rexsn 4685	. . . . . . . 8 ⊢ (∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅)
67		eqsn 4831	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
68	67	biimpar 476	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑦 = ∅) → 𝐴 = {∅})
69	66, 68	sylan2b 592	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → 𝐴 = {∅})
70	69	rexeqdv 3324	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
71	58, 70	mpbird 256	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
72	57, 71	jaodan 954	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
73	56, 72	sylan2b 592	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
74	55, 73	sylan2 591	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
75	1, 51, 74	syl2anc 582	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 394   ∨ wo 843   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947   ⊊ wpss 3948  ∅c0 4321  {csn 4627  ∪ cuni 4907   Or wor 5586  dom cdm 5675   [⊊] crpss 7714  Fincfn 8941  cardccrd 9932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rpss 7715  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-fin 8945  df-dju 9898  df-card 9936

This theorem is referenced by:  zornn0  10505  pgpfac1lem5  19990  lbsextlem4  20919  filssufilg  23635