Theorem zornn0g 10421

Description: Variant of Zorn's lemma zorng 10420 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 1138	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ≠ ∅)
2		simp1 1137	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → 𝐴 ∈ dom card)
3		snfi 8984	. . . . 5 ⊢ {∅} ∈ Fin
4		finnum 9866	. . . . 5 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
5	3, 4	ax-mp 5	. . . 4 ⊢ {∅} ∈ dom card
6		unnum 10113	. . . 4 ⊢ ((𝐴 ∈ dom card ∧ {∅} ∈ dom card) → (𝐴 ∪ {∅}) ∈ dom card)
7	2, 5, 6	sylancl 587	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → (𝐴 ∪ {∅}) ∈ dom card)
8		uncom 4099	. . . . . . . . 9 ⊢ (𝐴 ∪ {∅}) = ({∅} ∪ 𝐴)
9	8	sseq2i 3952	. . . . . . . 8 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ 𝑤 ⊆ ({∅} ∪ 𝐴))
10		ssundif 4428	. . . . . . . 8 ⊢ (𝑤 ⊆ ({∅} ∪ 𝐴) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
11	9, 10	bitri 275	. . . . . . 7 ⊢ (𝑤 ⊆ (𝐴 ∪ {∅}) ↔ (𝑤 ∖ {∅}) ⊆ 𝐴)
12		difss 4077	. . . . . . . . 9 ⊢ (𝑤 ∖ {∅}) ⊆ 𝑤
13		soss 5553	. . . . . . . . 9 ⊢ ((𝑤 ∖ {∅}) ⊆ 𝑤 → ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅})))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ( [⊊] Or 𝑤 → [⊊] Or (𝑤 ∖ {∅}))
15		ssdif0 4307	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {∅} ↔ (𝑤 ∖ {∅}) = ∅)
16		uni0b 4877	. . . . . . . . . . . . 13 ⊢ (∪ 𝑤 = ∅ ↔ 𝑤 ⊆ {∅})
17	16	biimpri 228	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ {∅} → ∪ 𝑤 = ∅)
18	17	eleq1d 2822	. . . . . . . . . . 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 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
22	21	difexi 5268	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {∅}) ∈ V
23		sseq1 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ⊆ 𝐴 ↔ (𝑤 ∖ {∅}) ⊆ 𝐴))
24		neeq1 2995	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → (𝑧 ≠ ∅ ↔ (𝑤 ∖ {∅}) ≠ ∅))
25		soeq2 5555	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ( [⊊] Or 𝑧 ↔ [⊊] Or (𝑤 ∖ {∅})))
26	23, 24, 25	3anbi123d 1439	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) ↔ ((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅ ∧ [⊊] Or (𝑤 ∖ {∅}))))
27		unieq 4862	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {∅}) → ∪ 𝑧 = ∪ (𝑤 ∖ {∅}))
28	27	eleq1d 2822	. . . . . . . . . . . . . . 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 3548	. . . . . . . . . . . . 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 1119	. . . . . . . . . . 11 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ (𝑤 ∖ {∅}) ≠ ∅) ∧ [⊊] Or (𝑤 ∖ {∅})) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
33	32	an32s 653	. . . . . . . . . 10 ⊢ ((((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or (𝑤 ∖ {∅})) ∧ (𝑤 ∖ {∅}) ≠ ∅) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ (𝑤 ∖ {∅}) ∈ 𝐴))
34		unidif0 5298	. . . . . . . . . . . 12 ⊢ ∪ (𝑤 ∖ {∅}) = ∪ 𝑤
35	34	eleq1i 2828	. . . . . . . . . . 11 ⊢ (∪ (𝑤 ∖ {∅}) ∈ 𝐴 ↔ ∪ 𝑤 ∈ 𝐴)
36		elun1 4123	. . . . . . . . . . 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 5243	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
40	39	snid 4607	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅}
41		elun2 4124	. . . . . . . . . . 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 594	. . . . . . 7 ⊢ (((𝑤 ∖ {∅}) ⊆ 𝐴 ∧ [⊊] Or 𝑤) → (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
46	11, 45	sylanb 582	. . . . . 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 1929	. . . 4 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
49	48	3ad2ant3 1136	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅})))
50		zorng 10420	. . 3 ⊢ (((𝐴 ∪ {∅}) ∈ dom card ∧ ∀𝑤((𝑤 ⊆ (𝐴 ∪ {∅}) ∧ [⊊] Or 𝑤) → ∪ 𝑤 ∈ (𝐴 ∪ {∅}))) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
51	7, 49, 50	syl2anc 585	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦)
52		ssun1 4119	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ {∅})
53		ssralv 3991	. . . . 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 3076	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦 → ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
56		rexun 4137	. . . 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 4031	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑥 ⊊ 𝑦 ↔ ∅ ⊊ 𝑦))
60		0pss 4388	. . . . . . . . . . . 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 3161	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
66	39, 65	rexsn 4627	. . . . . . 7 ⊢ (∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅)
67		eqsn 4773	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → (𝐴 = {∅} ↔ ∀𝑦 ∈ 𝐴 𝑦 = ∅))
68	67	biimpar 477	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑦 = ∅) → 𝐴 = {∅})
69	66, 68	sylan2b 595	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → 𝐴 = {∅})
70	58, 69	rexeqtrrdv 3301	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
71	57, 70	jaodan 960	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ∨ ∃𝑥 ∈ {∅}∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
72	56, 71	sylan2b 595	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
73	55, 72	sylan2 594	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ (𝐴 ∪ {∅})∀𝑦 ∈ (𝐴 ∪ {∅}) ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)
74	1, 51, 73	syl2anc 585	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 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890   ⊊ wpss 3891  ∅c0 4274  {csn 4568  ∪ cuni 4851   Or wor 5532  dom cdm 5625   [⊊] crpss 7670  Fincfn 8887  cardccrd 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-rpss 7671  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-fin 8891  df-dju 9819  df-card 9857

This theorem is referenced by:  zornn0  10424  pgpfac1lem5  20050  lbsextlem4  21154  filssufilg  23889