ltexprlem7 - Metamath Proof Explorer

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Theorem ltexprlem7 11083

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 11081	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → 𝐶 ∈ P)
3		ltaddpr 11075	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴<_P (𝐴 +_P 𝐶))
4		addclpr 11059	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 +_P 𝐶) ∈ P)
5		ltprord 11071	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ (𝐴 +_P 𝐶) ∈ P) → (𝐴<_P (𝐴 +_P 𝐶) ↔ 𝐴 ⊊ (𝐴 +_P 𝐶)))
6	4, 5	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴<_P (𝐴 +_P 𝐶) ↔ 𝐴 ⊊ (𝐴 +_P 𝐶)))
7	3, 6	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ⊊ (𝐴 +_P 𝐶))
8	7	pssssd 4099	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ⊆ (𝐴 +_P 𝐶))
9	8	sseld 3981	. . . . . . . . . . 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 11038	. . . . . . . . . . . 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 11032	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ P ∧ (𝑤 +_Q 𝑣) ∈ 𝐵) → (𝑤 +_Q 𝑣) ∈ Q)
16		addnqf 10989	. . . . . . . . . . . . . . . . . 18 ⊢ +_Q :(Q × Q)⟶Q
17	16	fdmi 6746	. . . . . . . . . . . . . . . . 17 ⊢ dom +_Q = (Q × Q)
18		0nnq 10965	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ ∈ Q
19	17, 18	ndmovrcl 7620	. . . . . . . . . . . . . . . 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 3483	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
23	22	prlem934 11074	. . . . . . . . . . . . . . . . . 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 11035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ Q) → (¬ 𝑤 ∈ 𝐴 → 𝑧 <_Q 𝑤))
26		ltexnq 11016	. . . . . . . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑧 ∈ 𝐴 ∧ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴)) → (𝑤 ∈ Q → (¬ 𝑤 ∈ 𝐴 → ∃𝑥(𝑧 +_Q 𝑥) = 𝑤)))
31		vex 3483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑧 ∈ V
32		vex 3483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑥 ∈ V
33		addcomnq 10992	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓)
34		addassnq 10999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 +_Q 𝑔) +_Q ℎ) = (𝑓 +_Q (𝑔 +_Q ℎ))
35	31, 22, 32, 33, 34	caov32 7661	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 +_Q 𝑣) +_Q 𝑥) = ((𝑧 +_Q 𝑥) +_Q 𝑣)
36		oveq1 7439	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑧 +_Q 𝑥) +_Q 𝑣) = (𝑤 +_Q 𝑣))
37	35, 36	eqtrid 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑧 +_Q 𝑣) +_Q 𝑥) = (𝑤 +_Q 𝑣))
38	37	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 +_Q 𝑣) ∈ V
41		eleq1 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → (𝑦 ∈ 𝐴 ↔ (𝑧 +_Q 𝑣) ∈ 𝐴))
42	41	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → (¬ 𝑦 ∈ 𝐴 ↔ ¬ (𝑧 +_Q 𝑣) ∈ 𝐴))
43		oveq1 7439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → (𝑦 +_Q 𝑥) = ((𝑧 +_Q 𝑣) +_Q 𝑥))
44	43	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑧 +_Q 𝑣) → ((𝑦 +_Q 𝑥) ∈ 𝐵 ↔ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵))
45	42, 44	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3605	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵) → ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵))
47	1	eqabri 2884	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ 𝐶 ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +_Q 𝑥) ∈ 𝐵))
48	46, 47	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑣) +_Q 𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐶)
49	39, 48	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((¬ (𝑧 +_Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +_Q 𝑥) = 𝑤 ∧ (𝑤 +_Q 𝑣) ∈ 𝐵)) → 𝑥 ∈ 𝐶)
50		df-plp 11024	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ +_P = (𝑥 ∈ P, 𝑤 ∈ P ↦ {𝑧 ∣ ∃𝑓 ∈ 𝑥 ∃𝑣 ∈ 𝑤 𝑧 = (𝑓 +_Q 𝑣)})
51		addclnq 10986	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ Q ∧ 𝑣 ∈ Q) → (𝑓 +_Q 𝑣) ∈ Q)
52	50, 51	genpprecl 11042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) → (𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶)))
53	49, 52	sylan2i 606	. . . . . . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 +_Q 𝑥) = 𝑤 → ((𝑧 +_Q 𝑥) ∈ (𝐴 +_P 𝐶) ↔ 𝑤 ∈ (𝐴 +_P 𝐶)))
57	56	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . 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 1932	. . . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . . . . 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 1932	. . . . . . . . . . 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 3988	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → 𝐵 ⊆ (𝐴 +_P 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∃wrex 3069 ⊆ wss 3950 ⊊ wpss 3951 class class class wbr 5142 × cxp 5682 (class class class)co 7432 Qcnq 10893 +_Q cplq 10896 <_Q cltq 10899 Pcnp 10900 +_P cpp 10902 <_P cltp 10904

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-ni 10913 df-pli 10914 df-mi 10915 df-lti 10916 df-plpq 10949 df-mpq 10950 df-ltpq 10951 df-enq 10952 df-nq 10953 df-erq 10954 df-plq 10955 df-mq 10956 df-1nq 10957 df-rq 10958 df-ltnq 10959 df-np 11022 df-plp 11024 df-ltp 11026

This theorem is referenced by: ltexpri 11084

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