Theorem zornn0g 10489

Description: Variant of Zorn's lemma zorng 10488 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 1153	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ≠ ∅)
2		simp1 1152	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ∈ dom card)
3		snfi 9040	. . . . 5 ⊢ {∅} ∈ Fin
4		finnum 9934	. . . . 5 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
5	3, 4	ax-mp 5	. . . 4 ⊢ {∅} ∈ dom card
6		unnum 10180	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
7	2, 5, 6	sylancl 597	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8		uncom 4120	. . . . . . . . 9 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
9	8	sseq2i 3974	. . . . . . . 8 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10		ssundif 4453	. . . . . . . 8 ⊢ (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
11	9, 10	bitri 278	. . . . . . 7 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12		difss 4098	. . . . . . . . 9 ⊢ (𝑤 ∖ {∅}) ⊆ 𝑤
13		soss 5590	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅})))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅}))
15		ssdif0 4329	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16		uni0b 4903	. . . . . . . . . . . . 13 ⊢ (∪ 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
17	16	biimpri 231	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ {∅} → ∪ 𝑤 = ∅)
18	17	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
19	15, 18	sylbir 238	. . . . . . . . . 10 ⊢ ((𝑤 ∖ {∅}) = ∅ → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
20	19	imbi2d 343	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) = ∅ → ((∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})) ↔ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∅ ∈ (𝐴 ∪ {∅}))))
21		vex 3467	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
22	21	difexi 5301	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {∅}) ∈ V
23		sseq1 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ⊆ 𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24		neeq1 3026	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25		soeq2 5592	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ( [⊊] Or 𝑧 ↔ [⊊] Or (𝑤 ∖ {∅})))
26	23, 24, 25	3anbi123d 1462	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅}))))
27		unieq 4887	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ∪ 𝑧 = ∪ (𝑤 ∖ {∅}))
28	27	eleq1d 2854	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (∪ 𝑧 ∈ 𝐴 ↔ ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
29	26, 28	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) ↔ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴)))
30	22, 29	spcv 3573	. . . . . . . . . . . . 13 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
31	30	com12 33	. . . . . . . . . . . 12 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
32	31	3expa 1134	. . . . . . . . . . 11 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
33	32	an32s 664	. . . . . . . . . 10 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
34		unidif0 5331	. . . . . . . . . . . 12 ⊢ ∪ (𝑤 ∖ {∅}) = ∪ 𝑤
35	34	eleq1i 2860	. . . . . . . . . . 11 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 ↔ ∪ 𝑤 ∈ 𝐴)
36		elun1 4143	. . . . . . . . . . 11 ⊢ (∪ 𝑤 ∈ 𝐴 → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))
37	35, 36	sylbi 220	. . . . . . . . . 10 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))
38	33, 37	syl6 36	. . . . . . . . 9 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
39		0ex 5272	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
40	39	snid 4633	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
41		elun2 4144	. . . . . . . . . . 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 3049	. . . . . . . 8 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
45	14, 44	sylan2 604	. . . . . . 7 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
46	11, 45	sylanb 592	. . . . . 6 ⊢ ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
47	46	com12 33	. . . . 5 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
48	47	alrimiv 1954	. . . 4 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
49	48	3ad2ant3 1151	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
50		zorng 10488	. . 3 ⊢ (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
51	7, 49, 50	syl2anc 595	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
52		ssun1 4139	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ {∅})
53		ssralv 4014	. . . . 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 3109	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
56		rexun 4157	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
57		simpr 489	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
58		simpr 489	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
59		psseq1 4052	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ ∅ ⊊ 𝑦))
60		0pss 4373	. . . . . . . . . . . 12 ⊢ (∅ ⊊ 𝑦 ↔ 𝑦 ≠ ∅)
61	59, 60	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ 𝑦 ≠ ∅))
62	61	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ 𝑦 ≠ ∅))
63		nne 2968	. . . . . . . . . 10 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
64	62, 63	bitrdi 290	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ 𝑦 = ∅))
65	64	ralbidv 3194	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
66	39, 65	rexsn 4653	. . . . . . 7 ⊢ (∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅)
67		eqsn 4799	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
68	67	biimpar 482	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑦 = ∅) → 𝐴 = {∅})
69	66, 68	sylan2b 605	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → 𝐴 = {∅})
70	58, 69	rexeqtrrdv 3334	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
71	57, 70	jaodan 972	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
72	56, 71	sylan2b 605	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
73	55, 72	sylan2 604	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
74	1, 51, 73	syl2anc 595	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 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101  ∀wal 1565   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913   ⊊ wpss 3914  ∅c0 4294  {csn 4594  ∪ cuni 4876   Or wor 5569  dom cdm 5662   [⊊] crpss 7720  Fincfn 8943  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rpss 7721  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-fin 8947  df-dju 9887  df-card 9925

This theorem is referenced by:  zornn0  10492  pgpfac1lem5  20151  lbsextlem4  21263  filssufilg  24037