ltexprlem7 - Metamath Proof Explorer

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Theorem ltexprlem7 10729

Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
ltexprlem.1	⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵)}

**Assertion**
Ref	Expression
ltexprlem7	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → 𝐵 ⊆ (𝐴 +_P 𝐶))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶 Allowed substitution hint: 𝐶(𝑦)

Proof of Theorem ltexprlem7 Dummy variables 𝑧 𝑤 𝑣 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . . . . 8 ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵)}
2	1	ltexprlem5 10727	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P)
3		ltaddpr 10721	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴<_P (𝐴 +_P 𝐶))
4		addclpr 10705	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 +_P 𝐶) ∈ P)
5		ltprord 10717	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ (𝐴 +_P 𝐶) ∈ P) → (𝐴<_P (𝐴 +_P 𝐶) ↔ 𝐴 ⊊ (𝐴 +_P 𝐶)))
6	4, 5	syldan 590	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴<_P (𝐴 +_P 𝐶) ↔ 𝐴 ⊊ (𝐴 +_P 𝐶)))
7	3, 6	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ⊊ (𝐴 +_P 𝐶))
8	7	pssssd 4028	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ⊆ (𝐴 +_P 𝐶))
9	8	sseld 3916	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ 𝐴 → 𝑤 ∈ (𝐴 +_P 𝐶)))
10	9	2a1d 26	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → (𝑤 ∈ 𝐴 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
11	10	com4r 94	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
12	11	expd 415	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐴 → (𝐴 ∈ P → (𝐶 ∈ P → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶))))))
13		prnmadd 10684	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ P ∧ 𝑤 ∈ 𝐵) → ∃𝑣(𝑤 +_Q 𝑣) ∈ 𝐵)
14	13	ex 412	. . . . . . . . . . 11 ⊢ (𝐵 ∈ P → (𝑤 ∈ 𝐵 → ∃𝑣(𝑤 +_Q 𝑣) ∈ 𝐵))
15		elprnq 10678	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ P ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → (𝑤 +_Q 𝑣) ∈ Q)
16		addnqf 10635	. . . . . . . . . . . . . . . . . 18 ⊢ +_Q :(Q × Q)⟶Q
17	16	fdmi 6596	. . . . . . . . . . . . . . . . 17 ⊢ dom +_Q = (Q × Q)
18		0nnq 10611	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ ∈ Q
19	17, 18	ndmovrcl 7436	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 +_Q 𝑣) ∈ Q → (𝑤 ∈ Q ∧ 𝑣 ∈ Q))
20	15, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ P ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → (𝑤 ∈ Q ∧ 𝑣 ∈ Q))
21	20	simpld 494	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → 𝑤 ∈ Q)
22		vex 3426	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
23	22	prlem934 10720	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ P → ∃𝑧 ∈ 𝐴 ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)
24	23	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ∃𝑧 ∈ 𝐴 ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)
25		prub 10681	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ Q) → (¬ 𝑤 ∈ 𝐴 → 𝑧 <_Q 𝑤))
26		ltexnq 10662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ Q → (𝑧 <_Q 𝑤 ↔ ∃𝑥(𝑧 +_Q 𝑥) = 𝑤))
27	26	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ Q) → (𝑧 <_Q 𝑤 ↔ ∃𝑥(𝑧 +_Q 𝑥) = 𝑤))
28	25, 27	sylibd 238	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ Q) → (¬ 𝑤 ∈ 𝐴 → ∃𝑥(𝑧 +_Q 𝑥) = 𝑤))
29	28	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ 𝐴) → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ∃𝑥(𝑧 +_Q 𝑥) = 𝑤)))
30	29	ad2ant2r 743	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ∃𝑥(𝑧 +_Q 𝑥) = 𝑤)))
31		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑧 ∈ V
32		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑥 ∈ V
33		addcomnq 10638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓)
34		addassnq 10645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 +_Q 𝑔) +_Q ℎ) = (𝑓 +_Q (𝑔 +_Q ℎ))
35	31, 22, 32, 33, 34	caov32 7477	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 +_Q 𝑣) +_Q 𝑥) = ((𝑧 +_Q 𝑥) +_Q 𝑣)
36		oveq1 7262	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑧 +_Q 𝑥) +_Q 𝑣) = (𝑤 +_Q 𝑣))
37	35, 36	eqtrid 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑧 +_Q 𝑣) +_Q 𝑥) = (𝑤 +_Q 𝑣))
38	37	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → (((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵 ↔ (𝑤 +_Q 𝑣) ∈ 𝐵))
39	38	biimpar 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵)
40		ovex 7288	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 +_Q 𝑣) ∈ V
41		eleq1 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → (𝑦 ∈ 𝐴 ↔ (𝑧 +_Q 𝑣) ∈ 𝐴))
42	41	notbid 317	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴))
43		oveq1 7262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → (𝑦 +_Q 𝑥) = ((𝑧 +_Q 𝑣) +_Q 𝑥))
44	43	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → ((𝑦 +_Q 𝑥) ∈ 𝐵 ↔ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵))
45	42, 44	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → ((¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵) ↔ (¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵)))
46	40, 45	spcev 3535	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵) → ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵))
47	1	abeq2i 2874	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ 𝐶 ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵))
48	46, 47	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐶)
49	39, 48	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵)) → 𝑥 ∈ 𝐶)
50		df-plp 10670	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ +_P = (𝑥 ∈ P, 𝑤 ∈ P ↦ {𝑧 ∣ ∃𝑓 ∈ 𝑥 ∃𝑣 ∈ 𝑤 𝑧 = (𝑓 +_Q 𝑣)})
51		addclnq 10632	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +_Q 𝑣) ∈ Q)
52	50, 51	genpprecl 10688	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶)))
53	49, 52	sylan2i 605	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑧 ∈ 𝐴 ∧ (¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵))) → (𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶)))
54	53	exp4d 433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑧 ∈ 𝐴 → (¬ (𝑧 +_Q 𝑣) ∈ 𝐴 → (((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → (𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶)))))
55	54	imp42 426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) ∧ ((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵)) → (𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶))
56		eleq1 2826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶) ↔ 𝑤 ∈ (𝐴 +_P 𝐶)))
57	56	ad2antrl 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) ∧ ((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵)) → ((𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶) ↔ 𝑤 ∈ (𝐴 +_P 𝐶)))
58	55, 57	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) ∧ ((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵)) → 𝑤 ∈ (𝐴 +_P 𝐶))
59	58	exp32 420	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) → ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑤 +_Q 𝑣) ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶))))
60	59	exlimdv 1937	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) → (∃𝑥(𝑧 +_Q 𝑥) = 𝑤 → ((𝑤 +_Q 𝑣) ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶))))
61	30, 60	syl6d 75	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ((𝑤 +_Q 𝑣) ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
62	24, 61	rexlimddv 3219	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ((𝑤 +_Q 𝑣) ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
63	62	com14 96	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 +_Q 𝑣) ∈ 𝐵 → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝑤 ∈ (𝐴 +_P 𝐶)))))
64	63	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝑤 ∈ (𝐴 +_P 𝐶)))))
65	21, 64	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝑤 ∈ (𝐴 +_P 𝐶))))
66	65	ex 412	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ P → ((𝑤 +_Q 𝑣) ∈ 𝐵 → (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝑤 ∈ (𝐴 +_P 𝐶)))))
67	66	exlimdv 1937	. . . . . . . . . . 11 ⊢ (𝐵 ∈ P → (∃𝑣(𝑤 +_Q 𝑣) ∈ 𝐵 → (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝑤 ∈ (𝐴 +_P 𝐶)))))
68	14, 67	syld 47	. . . . . . . . . 10 ⊢ (𝐵 ∈ P → (𝑤 ∈ 𝐵 → (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝑤 ∈ (𝐴 +_P 𝐶)))))
69	68	com4t 93	. . . . . . . . 9 ⊢ (¬ 𝑤 ∈ 𝐴 → ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
70	69	expd 415	. . . . . . . 8 ⊢ (¬ 𝑤 ∈ 𝐴 → (𝐴 ∈ P → (𝐶 ∈ P → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶))))))
71	12, 70	pm2.61i 182	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝐶 ∈ P → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
72	2, 71	syl5 34	. . . . . 6 ⊢ (𝐴 ∈ P → ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
73	72	expd 415	. . . . 5 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → (𝐵 ∈ P → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶))))))
74	73	com34 91	. . . 4 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶))))))
75	74	pm2.43d 53	. . 3 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝐴 ⊊ 𝐵 → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))))
76	75	imp31 417	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → (𝑤 ∈ 𝐵 → 𝑤 ∈ (𝐴 +_P 𝐶)))
77	76	ssrdv 3923	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → 𝐵 ⊆ (𝐴 +_P 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∃wex 1783 ∈ wcel 2108 {cab 2715 ∃wrex 3064 ⊆ wss 3883 ⊊ wpss 3884 class class class wbr 5070 × cxp 5578 (class class class)co 7255 Qcnq 10539 +_Q cplq 10542 <_Q cltq 10545 Pcnp 10546 +_P cpp 10548 <_P cltp 10550

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-omul 8272 df-er 8456 df-ni 10559 df-pli 10560 df-mi 10561 df-lti 10562 df-plpq 10595 df-mpq 10596 df-ltpq 10597 df-enq 10598 df-nq 10599 df-erq 10600 df-plq 10601 df-mq 10602 df-1nq 10603 df-rq 10604 df-ltnq 10605 df-np 10668 df-plp 10670 df-ltp 10672

This theorem is referenced by: ltexpri 10730

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