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Theorem usgrexmplef 27201

Description: Lemma for usgrexmpl 27205. (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 27200	. . 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 5300	. . . 4 ⊢ {0, 1} ∈ V
4		prex 5300	. . . 4 ⊢ {1, 2} ∈ V
5		prex 5300	. . . 4 ⊢ {2, 0} ∈ V
6		prex 5300	. . . 4 ⊢ {0, 3} ∈ V
7		s4f1o 14370	. . . 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 693	. . 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 6618	. 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 3402	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
13	12	elpr 4540	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} ↔ (𝑝 = {0, 1} ∨ 𝑝 = {1, 2}))
14		0nn0 11992	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
15		4nn0 11996	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
16		0re 10722	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
17		4re 11801	. . . . . . . . . . . . . . . . 17 ⊢ 4 ∈ ℝ
18		4pos 11824	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 4
19	16, 17, 18	ltleii 10842	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 4
20		elfz2nn0 13090	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...4) ↔ (0 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 0 ≤ 4))
21	14, 15, 19, 20	mpbir3an 1342	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ (0...4)
22		usgrexmplef.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (0...4)
23	21, 22	eleqtrri 2832	. . . . . . . . . . . . . 14 ⊢ 0 ∈ 𝑉
24		1nn0 11993	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
25		1re 10720	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
26		1lt4 11893	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 4
27	25, 17, 26	ltleii 10842	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ 4
28		elfz2nn0 13090	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (0...4) ↔ (1 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 1 ≤ 4))
29	24, 15, 27, 28	mpbir3an 1342	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (0...4)
30	29, 22	eleqtrri 2832	. . . . . . . . . . . . . 14 ⊢ 1 ∈ 𝑉
31		prelpwi 5307	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → {0, 1} ∈ 𝒫 𝑉)
32		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 1} ∈ 𝒫 𝑉))
33	31, 32	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉))
34	23, 30, 33	mp2an 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉)
35		fveq2 6675	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = (♯‘{0, 1}))
36		prhash2ex 13853	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 1}) = 2
37	35, 36	eqtrdi 2789	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = 2)
38	34, 37	jca 515	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
39		2nn0 11994	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
40		2re 11791	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
41		2lt4 11892	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 4
42	40, 17, 41	ltleii 10842	. . . . . . . . . . . . . . . 16 ⊢ 2 ≤ 4
43		elfz2nn0 13090	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 2 ≤ 4))
44	39, 15, 42, 43	mpbir3an 1342	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ (0...4)
45	44, 22	eleqtrri 2832	. . . . . . . . . . . . . 14 ⊢ 2 ∈ 𝑉
46		prelpwi 5307	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → {1, 2} ∈ 𝒫 𝑉)
47		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ↔ {1, 2} ∈ 𝒫 𝑉))
48	46, 47	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉))
49	30, 45, 48	mp2an 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉)
50		fveq2 6675	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = (♯‘{1, 2}))
51		1ne2 11925	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
52		1nn 11728	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
53		2nn 11790	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
54		hashprg 13849	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℕ ∧ 2 ∈ ℕ) → (1 ≠ 2 ↔ (♯‘{1, 2}) = 2))
55	52, 53, 54	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 ↔ (♯‘{1, 2}) = 2)
56	51, 55	mpbi 233	. . . . . . . . . . . . . 14 ⊢ (♯‘{1, 2}) = 2
57	50, 56	eqtrdi 2789	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = 2)
58	49, 57	jca 515	. . . . . . . . . . . 12 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
59	38, 58	jaoi 856	. . . . . . . . . . 11 ⊢ ((𝑝 = {0, 1} ∨ 𝑝 = {1, 2}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
60	13, 59	sylbi 220	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
61	12	elpr 4540	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} ↔ (𝑝 = {2, 0} ∨ 𝑝 = {0, 3}))
62		prelpwi 5307	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → {2, 0} ∈ 𝒫 𝑉)
63		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ↔ {2, 0} ∈ 𝒫 𝑉))
64	62, 63	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉))
65	45, 23, 64	mp2an 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉)
66		fveq2 6675	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = (♯‘{2, 0}))
67		2ne0 11821	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
68		2z 12096	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
69		0z 12074	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
70		hashprg 13849	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℤ ∧ 0 ∈ ℤ) → (2 ≠ 0 ↔ (♯‘{2, 0}) = 2))
71	68, 69, 70	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (2 ≠ 0 ↔ (♯‘{2, 0}) = 2)
72	67, 71	mpbi 233	. . . . . . . . . . . . . 14 ⊢ (♯‘{2, 0}) = 2
73	66, 72	eqtrdi 2789	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = 2)
74	65, 73	jca 515	. . . . . . . . . . . 12 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
75		3nn0 11995	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℕ₀
76		3re 11797	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
77		3lt4 11891	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 4
78	76, 17, 77	ltleii 10842	. . . . . . . . . . . . . . . 16 ⊢ 3 ≤ 4
79		elfz2nn0 13090	. . . . . . . . . . . . . . . 16 ⊢ (3 ∈ (0...4) ↔ (3 ∈ ℕ₀ ∧ 4 ∈ ℕ₀ ∧ 3 ≤ 4))
80	75, 15, 78, 79	mpbir3an 1342	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ (0...4)
81	80, 22	eleqtrri 2832	. . . . . . . . . . . . . 14 ⊢ 3 ∈ 𝑉
82		prelpwi 5307	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → {0, 3} ∈ 𝒫 𝑉)
83		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ↔ {0, 3} ∈ 𝒫 𝑉))
84	82, 83	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉))
85	23, 81, 84	mp2an 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉)
86		fveq2 6675	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = (♯‘{0, 3}))
87		3ne0 11823	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 0
88	87	necomi 2988	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 3
89		3z 12097	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
90		hashprg 13849	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0 ≠ 3 ↔ (♯‘{0, 3}) = 2))
91	69, 89, 90	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (0 ≠ 3 ↔ (♯‘{0, 3}) = 2)
92	88, 91	mpbi 233	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 3}) = 2
93	86, 92	eqtrdi 2789	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = 2)
94	85, 93	jca 515	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
95	74, 94	jaoi 856	. . . . . . . . . . 11 ⊢ ((𝑝 = {2, 0} ∨ 𝑝 = {0, 3}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
96	61, 95	sylbi 220	. . . . . . . . . 10 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
97	60, 96	jaoi 856	. . . . . . . . 9 ⊢ ((𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
98		elun 4040	. . . . . . . . 9 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}))
99		fveqeq2 6684	. . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → ((♯‘𝑒) = 2 ↔ (♯‘𝑝) = 2))
100	99	elrab 3588	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
101	97, 98, 100	3imtr4i 295	. . . . . . . 8 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
102	101	ssriv 3882	. . . . . . 7 ⊢ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
103	11, 102	sstrdi 3890	. . . . . 6 ⊢ (ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
104	103	anim2i 620	. . . . 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 6344	. . . . 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 6344	. . . . 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 295	. . . 4 ⊢ (𝐸:dom 𝐸⟶({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
108	107	anim1i 618	. . 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 6574	. . 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 6574	. . 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 295	. 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 209 ∧ wa 399 ∨ wo 846 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2113 ∃*wmo 2538 ≠ wne 2934 {crab 3057 Vcvv 3398 ∪ cun 3842 ⊆ wss 3844 𝒫 cpw 4489 {cpr 4519 class class class wbr 5031 dom cdm 5526 ran crn 5527 Fn wfn 6335 ⟶wf 6336 –1-1→wf1 6337 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7171 0cc0 10616 1c1 10617 ≤ cle 10755 ℕcn 11717 2c2 11772 3c3 11773 4c4 11774 ℕ₀cn0 11977 ℤcz 12063 ...cfz 12982 ♯chash 13783 ⟨“cs4 14295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-oadd 8136 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-dju 9404 df-card 9442 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-3 11781 df-4 11782 df-n0 11978 df-z 12064 df-uz 12326 df-fz 12983 df-fzo 13126 df-hash 13784 df-word 13957 df-concat 14013 df-s1 14040 df-s2 14300 df-s3 14301 df-s4 14302

This theorem is referenced by: usgrexmpl 27205

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