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Theorem suplocexprlemdisj 7719

Description: Lemma for suplocexpr 7724. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)

**Hypotheses**
Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<_P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<_P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<_P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<_P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩

**Assertion**
Ref	Expression
suplocexprlemdisj	⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵)))

Distinct variable groups: 𝑤,𝐴,𝑢 𝑥,𝐴,𝑦 𝑤,𝐵 𝜑,𝑞,𝑤 𝜑,𝑥,𝑦 𝑢,𝑞 Allowed substitution hints: 𝜑(𝑧,𝑢) 𝐴(𝑧,𝑞) 𝐵(𝑥,𝑦,𝑧,𝑢,𝑞)

Proof of Theorem suplocexprlemdisj Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 529	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) → 𝑞 ∈ ∪ (1^st “ 𝐴))
2		suplocexprlemell 7712	. . . . 5 ⊢ (𝑞 ∈ ∪ (1^st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑞 ∈ (1^st ‘𝑠))
3	1, 2	sylib 122	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1^st ‘𝑠))
4		simprr 531	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝑞 ∈ (1^st ‘𝑠))
5		simplrr 536	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝑞 ∈ (2^nd ‘𝐵))
6		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
7		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<_P 𝑥)
8		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<_P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<_P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<_P 𝑦)))
9	6, 7, 8	suplocexprlemss 7714	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
10	9	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝐴 ⊆ P)
11		suplocexpr.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = ⟨∪ (1^st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}⟩
12	11	suplocexprlem2b 7713	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ P → (2^nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢})
13	12	eleq2d 2247	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (𝑞 ∈ (2^nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}))
14	10, 13	syl 14	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → (𝑞 ∈ (2^nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢}))
15		breq2 4008	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <_Q 𝑢 ↔ 𝑤 <_Q 𝑞))
16	15	rexbidv 2478	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑞))
17	16	elrab 2894	. . . . . . . . . 10 ⊢ (𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑢} ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑞))
18	14, 17	bitrdi 196	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → (𝑞 ∈ (2^nd ‘𝐵) ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑞)))
19	5, 18	mpbid 147	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑞))
20	19	simprd 114	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → ∃𝑤 ∈ ∩ (2^nd “ 𝐴)𝑤 <_Q 𝑞)
21		simprr 531	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝑤 <_Q 𝑞)
22	10	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝐴 ⊆ P)
23		simplrl 535	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝑠 ∈ 𝐴)
24	22, 23	sseldd 3157	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝑠 ∈ P)
25		prop 7474	. . . . . . . . . 10 ⊢ (𝑠 ∈ P → ⟨(1^st ‘𝑠), (2^nd ‘𝑠)⟩ ∈ P)
26	24, 25	syl 14	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → ⟨(1^st ‘𝑠), (2^nd ‘𝑠)⟩ ∈ P)
27		eleq2 2241	. . . . . . . . . 10 ⊢ (𝑡 = (2^nd ‘𝑠) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2^nd ‘𝑠)))
28		simprl 529	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝑤 ∈ ∩ (2^nd “ 𝐴))
29		vex 2741	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
30	29	elint2 3852	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ∩ (2^nd “ 𝐴) ↔ ∀𝑡 ∈ (2^nd “ 𝐴)𝑤 ∈ 𝑡)
31	28, 30	sylib 122	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → ∀𝑡 ∈ (2^nd “ 𝐴)𝑤 ∈ 𝑡)
32		fo2nd 6159	. . . . . . . . . . . . . 14 ⊢ 2^nd :V–onto→V
33		fofun 5440	. . . . . . . . . . . . . 14 ⊢ (2^nd :V–onto→V → Fun 2^nd )
34	32, 33	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 2^nd
35		vex 2741	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
36		fof 5439	. . . . . . . . . . . . . . . 16 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
37	32, 36	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 2^nd :V⟶V
38	37	fdmi 5374	. . . . . . . . . . . . . 14 ⊢ dom 2^nd = V
39	35, 38	eleqtrri 2253	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ dom 2^nd
40		funfvima 5749	. . . . . . . . . . . . 13 ⊢ ((Fun 2^nd ∧ 𝑠 ∈ dom 2^nd ) → (𝑠 ∈ 𝐴 → (2^nd ‘𝑠) ∈ (2^nd “ 𝐴)))
41	34, 39, 40	mp2an 426	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (2^nd ‘𝑠) ∈ (2^nd “ 𝐴))
42	41	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → (2^nd ‘𝑠) ∈ (2^nd “ 𝐴))
43	42	adantr 276	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → (2^nd ‘𝑠) ∈ (2^nd “ 𝐴))
44	27, 31, 43	rspcdva 2847	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝑤 ∈ (2^nd ‘𝑠))
45		prcunqu 7484	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝑠), (2^nd ‘𝑠)⟩ ∈ P ∧ 𝑤 ∈ (2^nd ‘𝑠)) → (𝑤 <_Q 𝑞 → 𝑞 ∈ (2^nd ‘𝑠)))
46	26, 44, 45	syl2anc 411	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → (𝑤 <_Q 𝑞 → 𝑞 ∈ (2^nd ‘𝑠)))
47	21, 46	mpd 13	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2^nd “ 𝐴) ∧ 𝑤 <_Q 𝑞)) → 𝑞 ∈ (2^nd ‘𝑠))
48	20, 47	rexlimddv 2599	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝑞 ∈ (2^nd ‘𝑠))
49	4, 48	jca 306	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → (𝑞 ∈ (1^st ‘𝑠) ∧ 𝑞 ∈ (2^nd ‘𝑠)))
50		simprl 529	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝑠 ∈ 𝐴)
51	10, 50	sseldd 3157	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝑠 ∈ P)
52	51, 25	syl 14	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → ⟨(1^st ‘𝑠), (2^nd ‘𝑠)⟩ ∈ P)
53		simpllr 534	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → 𝑞 ∈ Q)
54		prdisj 7491	. . . . . 6 ⊢ ((⟨(1^st ‘𝑠), (2^nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1^st ‘𝑠) ∧ 𝑞 ∈ (2^nd ‘𝑠)))
55	52, 53, 54	syl2anc 411	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → ¬ (𝑞 ∈ (1^st ‘𝑠) ∧ 𝑞 ∈ (2^nd ‘𝑠)))
56	49, 55	pm2.21fal 1373	. . . 4 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1^st ‘𝑠))) → ⊥)
57	3, 56	rexlimddv 2599	. . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵))) → ⊥)
58	57	inegd 1372	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵)))
59	58	ralrimiva 2550	1 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1^st “ 𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ⊥wfal 1358 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 {crab 2459 Vcvv 2738 ⊆ wss 3130 ⟨cop 3596 ∪ cuni 3810 ∩ cint 3845 class class class wbr 4004 dom cdm 4627 “ cima 4630 Fun wfun 5211 ⟶wf 5213 –onto→wfo 5215 ‘cfv 5217 1^st c1st 6139 2^nd c2nd 6140 Qcnq 7279 <_Q cltq 7284 Pcnp 7290 <_P cltp 7294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-iinf 4588

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1st 6141 df-2nd 6142 df-qs 6541 df-ni 7303 df-nqqs 7347 df-ltnqqs 7352 df-inp 7465 df-iltp 7469

This theorem is referenced by: suplocexprlemex 7721

W3C validator