Theorem zornn0g 10524

Description: Variant of Zorn's lemma zorng 10523 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 1137	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ≠ ∅)
2		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ∈ dom card)
3		snfi 9062	. . . . 5 ⊢ {∅} ∈ Fin
4		finnum 9967	. . . . 5 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
5	3, 4	ax-mp 5	. . . 4 ⊢ {∅} ∈ dom card
6		unnum 10216	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
7	2, 5, 6	sylancl 586	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8		uncom 4138	. . . . . . . . 9 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
9	8	sseq2i 3993	. . . . . . . 8 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10		ssundif 4468	. . . . . . . 8 ⊢ (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
11	9, 10	bitri 275	. . . . . . 7 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12		difss 4116	. . . . . . . . 9 ⊢ (𝑤 ∖ {∅}) ⊆ 𝑤
13		soss 5586	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅})))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅}))
15		ssdif0 4346	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16		uni0b 4914	. . . . . . . . . . . . 13 ⊢ (∪ 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
17	16	biimpri 228	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ {∅} → ∪ 𝑤 = ∅)
18	17	eleq1d 2820	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} → (∪ 𝑤 ∈ (𝐴 ∪ {∅}) ↔ ∅ ∈ (𝐴 ∪ {∅})))
19	15, 18	sylbir 235	. . . . . . . . . 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 3468	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
22	21	difexi 5305	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {∅}) ∈ V
23		sseq1 3989	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ⊆ 𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24		neeq1 2995	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25		soeq2 5588	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ( [⊊] Or 𝑧 ↔ [⊊] Or (𝑤 ∖ {∅})))
26	23, 24, 25	3anbi123d 1438	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅}))))
27		unieq 4899	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ∪ 𝑧 = ∪ (𝑤 ∖ {∅}))
28	27	eleq1d 2820	. . . . . . . . . . . . . . 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 3589	. . . . . . . . . . . . 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 1118	. . . . . . . . . . 11 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
33	32	an32s 652	. . . . . . . . . 10 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
34		unidif0 5335	. . . . . . . . . . . 12 ⊢ ∪ (𝑤 ∖ {∅}) = ∪ 𝑤
35	34	eleq1i 2826	. . . . . . . . . . 11 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 ↔ ∪ 𝑤 ∈ 𝐴)
36		elun1 4162	. . . . . . . . . . 11 ⊢ (∪ 𝑤 ∈ 𝐴 → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))
37	35, 36	sylbi 217	. . . . . . . . . 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 5282	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
40	39	snid 4643	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
41		elun2 4163	. . . . . . . . . . 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 3018	. . . . . . . 8 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
45	14, 44	sylan2 593	. . . . . . 7 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
46	11, 45	sylanb 581	. . . . . 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 1927	. . . 4 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
49	48	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
50		zorng 10523	. . 3 ⊢ (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
51	7, 49, 50	syl2anc 584	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
52		ssun1 4158	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ {∅})
53		ssralv 4032	. . . . 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 3075	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
56		rexun 4176	. . . 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 4070	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ ∅ ⊊ 𝑦))
60		0pss 4427	. . . . . . . . . . . 12 ⊢ (∅ ⊊ 𝑦 ↔ 𝑦 ≠ ∅)
61	59, 60	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ 𝑦 ≠ ∅))
62	61	notbid 318	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ 𝑦 ≠ ∅))
63		nne 2937	. . . . . . . . . 10 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
64	62, 63	bitrdi 287	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (¬ 𝑥 ⊊ 𝑦 ↔ 𝑦 = ∅))
65	64	ralbidv 3164	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
66	39, 65	rexsn 4663	. . . . . . 7 ⊢ (∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅)
67		eqsn 4810	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
68	67	biimpar 477	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑦 = ∅) → 𝐴 = {∅})
69	66, 68	sylan2b 594	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → 𝐴 = {∅})
70	58, 69	rexeqtrrdv 3314	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
71	57, 70	jaodan 959	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
72	56, 71	sylan2b 594	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
73	55, 72	sylan2 593	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
74	1, 51, 73	syl2anc 584	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∖ cdif 3928   ∪ cun 3929   ⊆ wss 3931   ⊊ wpss 3932  ∅c0 4313  {csn 4606  ∪ cuni 4888   Or wor 5565  dom cdm 5659   [⊊] crpss 7721  Fincfn 8964  cardccrd 9954

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-rpss 7722  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8724  df-en 8965  df-dom 8966  df-fin 8968  df-dju 9920  df-card 9958

This theorem is referenced by:  zornn0  10527  pgpfac1lem5  20067  lbsextlem4  21127  filssufilg  23854