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Theorem usgrexmplef 29313

Description: Lemma for usgrexmpl 29317. (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 29312	. . 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 5381	. . . 4 ⊢ {0, 1} ∈ V
4		prex 5381	. . . 4 ⊢ {1, 2} ∈ V
5		prex 5381	. . . 4 ⊢ {2, 0} ∈ V
6		prex 5381	. . . 4 ⊢ {0, 3} ∈ V
7		s4f1o 14843	. . . 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 694	. . 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 6772	. 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 3443	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
13	12	elpr 4604	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} ↔ (𝑝 = {0, 1} ∨ 𝑝 = {1, 2}))
14		0nn0 12418	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
15		4nn0 12422	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
16		0re 11136	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
17		4re 12231	. . . . . . . . . . . . . . . . 17 ⊢ 4 ∈ ℝ
18		4pos 12254	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 4
19	16, 17, 18	ltleii 11258	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 4
20		elfz2nn0 13536	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...4) ↔ (0 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 0 ≤ 4))
21	14, 15, 19, 20	mpbir3an 1343	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...4)
22		usgrexmplef.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (0...4)
23	21, 22	eleqtrri 2834	. . . . . . . . . . . . . 14 ⊢ 0 ∈ 𝑉
24		1nn0 12419	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
25		1re 11134	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
26		1lt4 12318	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 4
27	25, 17, 26	ltleii 11258	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ 4
28		elfz2nn0 13536	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (0...4) ↔ (1 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 1 ≤ 4))
29	24, 15, 27, 28	mpbir3an 1343	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (0...4)
30	29, 22	eleqtrri 2834	. . . . . . . . . . . . . 14 ⊢ 1 ∈ 𝑉
31		prelpwi 5394	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → {0, 1} ∈ 𝒫 𝑉)
32		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 1} ∈ 𝒫 𝑉))
33	31, 32	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉))
34	23, 30, 33	mp2an 693	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉)
35		fveq2 6833	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = (♯‘{0, 1}))
36		prhash2ex 14324	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 1}) = 2
37	35, 36	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = 2)
38	34, 37	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
39		2nn0 12420	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
40		2re 12221	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
41		2lt4 12317	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 4
42	40, 17, 41	ltleii 11258	. . . . . . . . . . . . . . . 16 ⊢ 2 ≤ 4
43		elfz2nn0 13536	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 2 ≤ 4))
44	39, 15, 42, 43	mpbir3an 1343	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (0...4)
45	44, 22	eleqtrri 2834	. . . . . . . . . . . . . 14 ⊢ 2 ∈ 𝑉
46		prelpwi 5394	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → {1, 2} ∈ 𝒫 𝑉)
47		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ↔ {1, 2} ∈ 𝒫 𝑉))
48	46, 47	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉))
49	30, 45, 48	mp2an 693	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉)
50		fveq2 6833	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = (♯‘{1, 2}))
51		1ne2 12350	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
52		1nn 12158	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
53		2nn 12220	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
54		hashprg 14320	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → (1 ≠ 2 ↔ (♯‘{1, 2}) = 2))
55	52, 53, 54	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 ↔ (♯‘{1, 2}) = 2)
56	51, 55	mpbi 230	. . . . . . . . . . . . . 14 ⊢ (♯‘{1, 2}) = 2
57	50, 56	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = 2)
58	49, 57	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
59	38, 58	jaoi 858	. . . . . . . . . . 11 ⊢ ((𝑝 = {0, 1} ∨ 𝑝 = {1, 2}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
60	13, 59	sylbi 217	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
61	12	elpr 4604	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} ↔ (𝑝 = {2, 0} ∨ 𝑝 = {0, 3}))
62		prelpwi 5394	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → {2, 0} ∈ 𝒫 𝑉)
63		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ↔ {2, 0} ∈ 𝒫 𝑉))
64	62, 63	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉))
65	45, 23, 64	mp2an 693	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉)
66		fveq2 6833	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = (♯‘{2, 0}))
67		2ne0 12251	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
68		2z 12525	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
69		0z 12501	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
70		hashprg 14320	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℤ ∧ 0 ∈ ℤ) → (2 ≠ 0 ↔ (♯‘{2, 0}) = 2))
71	68, 69, 70	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (2 ≠ 0 ↔ (♯‘{2, 0}) = 2)
72	67, 71	mpbi 230	. . . . . . . . . . . . . 14 ⊢ (♯‘{2, 0}) = 2
73	66, 72	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = 2)
74	65, 73	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
75		3nn0 12421	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℕ₀
76		3re 12227	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
77		3lt4 12316	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 4
78	76, 17, 77	ltleii 11258	. . . . . . . . . . . . . . . 16 ⊢ 3 ≤ 4
79		elfz2nn0 13536	. . . . . . . . . . . . . . . 16 ⊢ (3 ∈ (0...4) ↔ (3 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 3 ≤ 4))
80	75, 15, 78, 79	mpbir3an 1343	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ (0...4)
81	80, 22	eleqtrri 2834	. . . . . . . . . . . . . 14 ⊢ 3 ∈ 𝑉
82		prelpwi 5394	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → {0, 3} ∈ 𝒫 𝑉)
83		eleq1 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 3} ∈ 𝒫 𝑉))
84	82, 83	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉))
85	23, 81, 84	mp2an 693	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉)
86		fveq2 6833	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = (♯‘{0, 3}))
87		3ne0 12253	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 0
88	87	necomi 2985	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 3
89		3z 12526	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
90		hashprg 14320	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0 ≠ 3 ↔ (♯‘{0, 3}) = 2))
91	69, 89, 90	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (0 ≠ 3 ↔ (♯‘{0, 3}) = 2)
92	88, 91	mpbi 230	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 3}) = 2
93	86, 92	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = 2)
94	85, 93	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
95	74, 94	jaoi 858	. . . . . . . . . . 11 ⊢ ((𝑝 = {2, 0} ∨ 𝑝 = {0, 3}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
96	61, 95	sylbi 217	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
97	60, 96	jaoi 858	. . . . . . . . 9 ⊢ ((𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
98		elun 4104	. . . . . . . . 9 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}))
99		fveqeq2 6842	. . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → ((♯‘𝑒) = 2 ↔ (♯‘𝑝) = 2))
100	99	elrab 3645	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
101	97, 98, 100	3imtr4i 292	. . . . . . . 8 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
102	101	ssriv 3936	. . . . . . 7 ⊢ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
103	11, 102	sstrdi 3945	. . . . . 6 ⊢ (ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
104	103	anim2i 618	. . . . 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 6495	. . . . 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 6495	. . . . 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 292	. . . 4 ⊢ (𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
108	107	anim1i 616	. . 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 6728	. . 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 6728	. . 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 292	. 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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ∃*wmo 2536 ≠ wne 2931 {crab 3398 Vcvv 3439 ∪ cun 3898 ⊆ wss 3900 𝒫 cpw 4553 {cpr 4581 class class class wbr 5097 dom cdm 5623 ran crn 5624 Fn wfn 6486 ⟶wf 6487 –1-1→wf1 6488 –1-1-onto→wf1o 6490 ‘cfv 6491 (class class class)co 7358 0cc0 11028 1c1 11029 ≤ cle 11169 ℕcn 12147 2c2 12202 3c3 12203 4c4 12204 ℕ₀cn0 12403 ℤcz 12490 ...cfz 13425 ♯chash 14255 ⟨“cs4 14768

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-oadd 8401 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-n0 12404 df-z 12491 df-uz 12754 df-fz 13426 df-fzo 13573 df-hash 14256 df-word 14439 df-concat 14496 df-s1 14522 df-s2 14773 df-s3 14774 df-s4 14775

This theorem is referenced by: usgrexmpl 29317

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