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Theorem usgrexmplef 26151

Description: Lemma for usgrexmpl 26155. (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 26150	. . 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 4909	. . . 4 ⊢ {0, 1} ∈ V
4		prex 4909	. . . 4 ⊢ {1, 2} ∈ V
5		prex 4909	. . . 4 ⊢ {2, 0} ∈ V
6		prex 4909	. . . 4 ⊢ {0, 3} ∈ V
7		s4f1o 13663	. . . 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 709	. . 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 6136	. 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 3203	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
13	12	elpr 4198	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} ↔ (𝑝 = {0, 1} ∨ 𝑝 = {1, 2}))
14		0nn0 11307	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
15		4nn0 11311	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
16		0re 10040	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
17		4re 11097	. . . . . . . . . . . . . . . . 17 ⊢ 4 ∈ ℝ
18		4pos 11116	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 4
19	16, 17, 18	ltleii 10160	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 4
20		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...4) ↔ (0 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 0 ≤ 4))
21	14, 15, 19, 20	mpbir3an 1244	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...4)
22		usgrexmplef.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (0...4)
23	21, 22	eleqtrri 2700	. . . . . . . . . . . . . 14 ⊢ 0 ∈ 𝑉
24		1nn0 11308	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
25		1re 10039	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
26		1lt4 11199	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 4
27	25, 17, 26	ltleii 10160	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ 4
28		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (0...4) ↔ (1 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 1 ≤ 4))
29	24, 15, 27, 28	mpbir3an 1244	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (0...4)
30	29, 22	eleqtrri 2700	. . . . . . . . . . . . . 14 ⊢ 1 ∈ 𝑉
31		prelpwi 4915	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → {0, 1} ∈ 𝒫 𝑉)
32		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 1} ∈ 𝒫 𝑉))
33	31, 32	syl5ibrcom 237	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉))
34	23, 30, 33	mp2an 708	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉)
35		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 1} → (#‘𝑝) = (#‘{0, 1}))
36		prhash2ex 13187	. . . . . . . . . . . . . 14 ⊢ (#‘{0, 1}) = 2
37	35, 36	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → (#‘𝑝) = 2)
38	34, 37	jca 554	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
39		2nn0 11309	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
40		2re 11090	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
41		2lt4 11198	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 4
42	40, 17, 41	ltleii 10160	. . . . . . . . . . . . . . . 16 ⊢ 2 ≤ 4
43		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 2 ≤ 4))
44	39, 15, 42, 43	mpbir3an 1244	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (0...4)
45	44, 22	eleqtrri 2700	. . . . . . . . . . . . . 14 ⊢ 2 ∈ 𝑉
46		prelpwi 4915	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → {1, 2} ∈ 𝒫 𝑉)
47		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ↔ {1, 2} ∈ 𝒫 𝑉))
48	46, 47	syl5ibrcom 237	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉))
49	30, 45, 48	mp2an 708	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉)
50		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {1, 2} → (#‘𝑝) = (#‘{1, 2}))
51		1ne2 11240	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
52		1nn 11031	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
53		2nn 11185	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
54		hashprg 13182	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → (1 ≠ 2 ↔ (#‘{1, 2}) = 2))
55	52, 53, 54	mp2an 708	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 ↔ (#‘{1, 2}) = 2)
56	51, 55	mpbi 220	. . . . . . . . . . . . . 14 ⊢ (#‘{1, 2}) = 2
57	50, 56	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → (#‘𝑝) = 2)
58	49, 57	jca 554	. . . . . . . . . . . 12 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
59	38, 58	jaoi 394	. . . . . . . . . . 11 ⊢ ((𝑝 = {0, 1} ∨ 𝑝 = {1, 2}) → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
60	13, 59	sylbi 207	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
61	12	elpr 4198	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} ↔ (𝑝 = {2, 0} ∨ 𝑝 = {0, 3}))
62		prelpwi 4915	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → {2, 0} ∈ 𝒫 𝑉)
63		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ↔ {2, 0} ∈ 𝒫 𝑉))
64	62, 63	syl5ibrcom 237	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉))
65	45, 23, 64	mp2an 708	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉)
66		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {2, 0} → (#‘𝑝) = (#‘{2, 0}))
67		2ne0 11113	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
68		2z 11409	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
69		0z 11388	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
70		hashprg 13182	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℤ ∧ 0 ∈ ℤ) → (2 ≠ 0 ↔ (#‘{2, 0}) = 2))
71	68, 69, 70	mp2an 708	. . . . . . . . . . . . . . 15 ⊢ (2 ≠ 0 ↔ (#‘{2, 0}) = 2)
72	67, 71	mpbi 220	. . . . . . . . . . . . . 14 ⊢ (#‘{2, 0}) = 2
73	66, 72	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → (#‘𝑝) = 2)
74	65, 73	jca 554	. . . . . . . . . . . 12 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
75		3nn0 11310	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℕ₀
76		3re 11094	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
77		3lt4 11197	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 4
78	76, 17, 77	ltleii 10160	. . . . . . . . . . . . . . . 16 ⊢ 3 ≤ 4
79		elfz2nn0 12431	. . . . . . . . . . . . . . . 16 ⊢ (3 ∈ (0...4) ↔ (3 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 3 ≤ 4))
80	75, 15, 78, 79	mpbir3an 1244	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ (0...4)
81	80, 22	eleqtrri 2700	. . . . . . . . . . . . . 14 ⊢ 3 ∈ 𝑉
82		prelpwi 4915	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → {0, 3} ∈ 𝒫 𝑉)
83		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 3} ∈ 𝒫 𝑉))
84	82, 83	syl5ibrcom 237	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉))
85	23, 81, 84	mp2an 708	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉)
86		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 3} → (#‘𝑝) = (#‘{0, 3}))
87		3ne0 11115	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 0
88	87	necomi 2848	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 3
89		3z 11410	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
90		hashprg 13182	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0 ≠ 3 ↔ (#‘{0, 3}) = 2))
91	69, 89, 90	mp2an 708	. . . . . . . . . . . . . . 15 ⊢ (0 ≠ 3 ↔ (#‘{0, 3}) = 2)
92	88, 91	mpbi 220	. . . . . . . . . . . . . 14 ⊢ (#‘{0, 3}) = 2
93	86, 92	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → (#‘𝑝) = 2)
94	85, 93	jca 554	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
95	74, 94	jaoi 394	. . . . . . . . . . 11 ⊢ ((𝑝 = {2, 0} ∨ 𝑝 = {0, 3}) → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
96	61, 95	sylbi 207	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
97	60, 96	jaoi 394	. . . . . . . . 9 ⊢ ((𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}) → (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
98		elun 3753	. . . . . . . . 9 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}))
99		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑝 → (#‘𝑒) = (#‘𝑝))
100	99	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → ((#‘𝑒) = 2 ↔ (#‘𝑝) = 2))
101	100	elrab 3363	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (#‘𝑝) = 2))
102	97, 98, 101	3imtr4i 281	. . . . . . . 8 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
103	102	ssriv 3607	. . . . . . 7 ⊢ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}
104	11, 103	syl6ss 3615	. . . . . 6 ⊢ (ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
105	104	anim2i 593	. . . . 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}))
106		df-f 5892	. . . . 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}})))
107		df-f 5892	. . . . 5 ⊢ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ↔ (𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}))
108	105, 106, 107	3imtr4i 281	. . . 4 ⊢ (𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
109	108	anim1i 592	. . 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))
110		dff12 6100	. . 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))
111		dff12 6100	. . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ↔ (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ ∀𝑥∃*𝑦 𝑦𝐸𝑥))
112	109, 110, 111	3imtr4i 281	. 2 ⊢ (𝐸:dom 𝐸–1-1→({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
113	9, 10, 112	mp2b 10	1 ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∃*wmo 2471 ≠ wne 2794 {crab 2916 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 {cpr 4179 class class class wbr 4653 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 ≤ cle 10075 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 #chash 13117 ⟨“cs4 13588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-s4 13595

This theorem is referenced by: usgrexmpl 26155

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