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Theorem usgrexmplef 29462

Description: Lemma for usgrexmpl 29466. (Contributed by Alexander van der Vekens, 15-Aug-2017.)

**Hypotheses**
Ref	Expression
usgrexmplef.v	⊢ 𝑉 = (0...4)
usgrexmplef.e	⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩

**Assertion**
Ref	Expression
usgrexmplef	⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}

Distinct variable groups: 𝑒,𝐸 𝑒,𝑉

Proof of Theorem usgrexmplef Dummy variables 𝑥 𝑦 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrexmpldifpr 29461	. . 3 ⊢ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3}))
2		usgrexmplef.e	. . 3 ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩
3		prex 5397	. . . 4 ⊢ {0, 1} ∈ V
4		prex 5397	. . . 4 ⊢ {1, 2} ∈ V
5		prex 5397	. . . 4 ⊢ {2, 0} ∈ V
6		prex 5397	. . . 4 ⊢ {0, 3} ∈ V
7		s4f1o 14933	. . . 4 ⊢ ((({0, 1} ∈ V ∧ {1, 2} ∈ V) ∧ ({2, 0} ∈ V ∧ {0, 3} ∈ V)) → ((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3})) → (𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩ → 𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}))))
8	3, 4, 5, 6, 7	mp4an 703	. . 3 ⊢ ((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3})) → (𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩ → 𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})))
9	1, 2, 8	mp2 9	. 2 ⊢ 𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})
10		f1of1 6807	. 2 ⊢ (𝐸:dom 𝐸–1-1-onto→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸–1-1→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}))
11		id 22	. . . . . . 7 ⊢ (ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}))
12		vex 3460	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
13	12	elpr 4609	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} ↔ (𝑝 = {0, 1} ∨ 𝑝 = {1, 2}))
14		0nn0 12498	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
15		4nn0 12502	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
16		0re 11185	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
17		4re 12304	. . . . . . . . . . . . . . . . 17 ⊢ 4 ∈ ℝ
18		4pos 12330	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 4
19	16, 17, 18	ltleii 11308	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 4
20		elfz2nn0 13625	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...4) ↔ (0 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 0 ≤ 4))
21	14, 15, 19, 20	mpbir3an 1356	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...4)
22		usgrexmplef.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (0...4)
23	21, 22	eleqtrri 2863	. . . . . . . . . . . . . 14 ⊢ 0 ∈ 𝑉
24		1nn0 12499	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
25		1re 11183	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
26		1lt4 12398	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 4
27	25, 17, 26	ltleii 11308	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ 4
28		elfz2nn0 13625	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (0...4) ↔ (1 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 1 ≤ 4))
29	24, 15, 27, 28	mpbir3an 1356	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (0...4)
30	29, 22	eleqtrri 2863	. . . . . . . . . . . . . 14 ⊢ 1 ∈ 𝑉
31		prelpwi 5416	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → {0, 1} ∈ 𝒫 𝑉)
32		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 1} ∈ 𝒫 𝑉))
33	31, 32	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉))
34	23, 30, 33	mp2an 702	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉)
35		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = (♯‘{0, 1}))
36		prhash2ex 14414	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 1}) = 2
37	35, 36	eqtrdi 2815	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = 2)
38	34, 37	jca 519	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
39		2nn0 12500	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
40		2re 12294	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
41		2lt4 12397	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 4
42	40, 17, 41	ltleii 11308	. . . . . . . . . . . . . . . 16 ⊢ 2 ≤ 4
43		elfz2nn0 13625	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 2 ≤ 4))
44	39, 15, 42, 43	mpbir3an 1356	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (0...4)
45	44, 22	eleqtrri 2863	. . . . . . . . . . . . . 14 ⊢ 2 ∈ 𝑉
46		prelpwi 5416	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → {1, 2} ∈ 𝒫 𝑉)
47		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ↔ {1, 2} ∈ 𝒫 𝑉))
48	46, 47	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉))
49	30, 45, 48	mp2an 702	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉)
50		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = (♯‘{1, 2}))
51		1ne2 12430	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
52		1nn 12223	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
53		2nn 12293	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
54		hashprg 14410	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → (1 ≠ 2 ↔ (♯‘{1, 2}) = 2))
55	52, 53, 54	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 ↔ (♯‘{1, 2}) = 2)
56	51, 55	mpbi 232	. . . . . . . . . . . . . 14 ⊢ (♯‘{1, 2}) = 2
57	50, 56	eqtrdi 2815	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = 2)
58	49, 57	jca 519	. . . . . . . . . . . 12 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
59	38, 58	jaoi 868	. . . . . . . . . . 11 ⊢ ((𝑝 = {0, 1} ∨ 𝑝 = {1, 2}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
60	13, 59	sylbi 219	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
61	12	elpr 4609	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} ↔ (𝑝 = {2, 0} ∨ 𝑝 = {0, 3}))
62		prelpwi 5416	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → {2, 0} ∈ 𝒫 𝑉)
63		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ↔ {2, 0} ∈ 𝒫 𝑉))
64	62, 63	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉))
65	45, 23, 64	mp2an 702	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉)
66		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = (♯‘{2, 0}))
67		2ne0 12326	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
68		2z 12605	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
69		0z 12581	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
70		hashprg 14410	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℤ ∧ 0 ∈ ℤ) → (2 ≠ 0 ↔ (♯‘{2, 0}) = 2))
71	68, 69, 70	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (2 ≠ 0 ↔ (♯‘{2, 0}) = 2)
72	67, 71	mpbi 232	. . . . . . . . . . . . . 14 ⊢ (♯‘{2, 0}) = 2
73	66, 72	eqtrdi 2815	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = 2)
74	65, 73	jca 519	. . . . . . . . . . . 12 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
75		3nn0 12501	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℕ₀
76		3re 12300	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
77		3lt4 12396	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 4
78	76, 17, 77	ltleii 11308	. . . . . . . . . . . . . . . 16 ⊢ 3 ≤ 4
79		elfz2nn0 13625	. . . . . . . . . . . . . . . 16 ⊢ (3 ∈ (0...4) ↔ (3 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 3 ≤ 4))
80	75, 15, 78, 79	mpbir3an 1356	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ (0...4)
81	80, 22	eleqtrri 2863	. . . . . . . . . . . . . 14 ⊢ 3 ∈ 𝑉
82		prelpwi 5416	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → {0, 3} ∈ 𝒫 𝑉)
83		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 3} ∈ 𝒫 𝑉))
84	82, 83	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉))
85	23, 81, 84	mp2an 702	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉)
86		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = (♯‘{0, 3}))
87		3ne0 12329	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 0
88	87	necomi 3013	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 3
89		3z 12606	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
90		hashprg 14410	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0 ≠ 3 ↔ (♯‘{0, 3}) = 2))
91	69, 89, 90	mp2an 702	. . . . . . . . . . . . . . 15 ⊢ (0 ≠ 3 ↔ (♯‘{0, 3}) = 2)
92	88, 91	mpbi 232	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 3}) = 2
93	86, 92	eqtrdi 2815	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = 2)
94	85, 93	jca 519	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
95	74, 94	jaoi 868	. . . . . . . . . . 11 ⊢ ((𝑝 = {2, 0} ∨ 𝑝 = {0, 3}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
96	61, 95	sylbi 219	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
97	60, 96	jaoi 868	. . . . . . . . 9 ⊢ ((𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
98		elun 4108	. . . . . . . . 9 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}))
99		fveqeq2 6878	. . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → ((♯‘𝑒) = 2 ↔ (♯‘𝑝) = 2))
100	99	elrab 3652	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
101	97, 98, 100	3imtr4i 294	. . . . . . . 8 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
102	101	ssriv 3942	. . . . . . 7 ⊢ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
103	11, 102	sstrdi 3950	. . . . . 6 ⊢ (ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
104	103	anim2i 626	. . . . 5 ⊢ ((𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})) → (𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}))
105		df-f 6527	. . . . 5 ⊢ (𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})))
106		df-f 6527	. . . . 5 ⊢ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ (𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}))
107	104, 105, 106	3imtr4i 294	. . . 4 ⊢ (𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
108	107	anim1i 624	. . 3 ⊢ ((𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ∧ ∀𝑥∃𝑦 𝑦𝐸𝑥) → (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ∧ ∀𝑥∃𝑦 𝑦𝐸𝑥))
109		dff12 6761	. . 3 ⊢ (𝐸:dom 𝐸–1-1→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ∧ ∀𝑥∃*𝑦 𝑦𝐸𝑥))
110		dff12 6761	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ∧ ∀𝑥∃*𝑦 𝑦𝐸𝑥))
111	108, 109, 110	3imtr4i 294	. 2 ⊢ (𝐸:dom 𝐸–1-1→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
112	9, 10, 111	mp2b 10	1 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1099 ∀wal 1560 = wceq 1562 ∈ wcel 2144 ∃*wmo 2566 ≠ wne 2959 {crab 3416 Vcvv 3456 ∪ cun 3904 ⊆ wss 3906 𝒫 cpw 4557 {cpr 4586 class class class wbr 5102 dom cdm 5649 ran crn 5650 Fn wfn 6518 ⟶wf 6519 –1-1→wf1 6520 –1-1-onto→wf1o 6522 ‘cfv 6523 (class class class)co 7398 0cc0 11075 1c1 11076 ≤ cle 11219 ℕcn 12212 2c2 12274 3c3 12275 4c4 12276 ℕ₀cn0 12483 ℤcz 12570 ...cfz 13514 ♯chash 14345 ⟨“cs4 14858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-oadd 8443 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-dju 9861 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-n0 12484 df-z 12571 df-uz 12842 df-fz 13515 df-fzo 13662 df-hash 14346 df-word 14529 df-concat 14586 df-s1 14612 df-s2 14863 df-s3 14864 df-s4 14865

This theorem is referenced by: usgrexmpl 29466

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