usgrexmplef - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > usgrexmplef		Structured version Visualization version GIF version

Theorem usgrexmplef 29186

Description: Lemma for usgrexmpl 29190. (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 29185	. . 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 5392	. . . 4 ⊢ {0, 1} ∈ V
4		prex 5392	. . . 4 ⊢ {1, 2} ∈ V
5		prex 5392	. . . 4 ⊢ {2, 0} ∈ V
6		prex 5392	. . . 4 ⊢ {0, 3} ∈ V
7		s4f1o 14884	. . . 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 6799	. 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 3451	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
13	12	elpr 4614	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{0, 1}, {1, 2}} ↔ (𝑝 = {0, 1} ∨ 𝑝 = {1, 2}))
14		0nn0 12457	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
15		4nn0 12461	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
16		0re 11176	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
17		4re 12270	. . . . . . . . . . . . . . . . 17 ⊢ 4 ∈ ℝ
18		4pos 12293	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 4
19	16, 17, 18	ltleii 11297	. . . . . . . . . . . . . . . 16 ⊢ 0 ≤ 4
20		elfz2nn0 13579	. . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 14 ⊢ 0 ∈ 𝑉
24		1nn0 12458	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
25		1re 11174	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
26		1lt4 12357	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 4
27	25, 17, 26	ltleii 11297	. . . . . . . . . . . . . . . 16 ⊢ 1 ≤ 4
28		elfz2nn0 13579	. . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 14 ⊢ 1 ∈ 𝑉
31		prelpwi 5407	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 1 ∈ 𝑉) → {0, 1} ∈ 𝒫 𝑉)
32		eleq1 2816	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → 𝑝 ∈ 𝒫 𝑉)
35		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = (♯‘{0, 1}))
36		prhash2ex 14364	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 1}) = 2
37	35, 36	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 1} → (♯‘𝑝) = 2)
38	34, 37	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 1} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
39		2nn0 12459	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
40		2re 12260	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
41		2lt4 12356	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 4
42	40, 17, 41	ltleii 11297	. . . . . . . . . . . . . . . 16 ⊢ 2 ≤ 4
43		elfz2nn0 13579	. . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 14 ⊢ 2 ∈ 𝑉
46		prelpwi 5407	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ 𝑉 ∧ 2 ∈ 𝑉) → {1, 2} ∈ 𝒫 𝑉)
47		eleq1 2816	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → 𝑝 ∈ 𝒫 𝑉)
50		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = (♯‘{1, 2}))
51		1ne2 12389	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
52		1nn 12197	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ
53		2nn 12259	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
54		hashprg 14360	. . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . 14 ⊢ (♯‘{1, 2}) = 2
57	50, 56	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑝 = {1, 2} → (♯‘𝑝) = 2)
58	49, 57	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {1, 2} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
59	38, 58	jaoi 857	. . . . . . . . . . 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 4614	. . . . . . . . . . 11 ⊢ (𝑝 ∈ {{2, 0}, {0, 3}} ↔ (𝑝 = {2, 0} ∨ 𝑝 = {0, 3}))
62		prelpwi 5407	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ 𝑉 ∧ 0 ∈ 𝑉) → {2, 0} ∈ 𝒫 𝑉)
63		eleq1 2816	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → 𝑝 ∈ 𝒫 𝑉)
66		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = (♯‘{2, 0}))
67		2ne0 12290	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
68		2z 12565	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
69		0z 12540	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
70		hashprg 14360	. . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . 14 ⊢ (♯‘{2, 0}) = 2
73	66, 72	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑝 = {2, 0} → (♯‘𝑝) = 2)
74	65, 73	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {2, 0} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
75		3nn0 12460	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℕ₀
76		3re 12266	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
77		3lt4 12355	. . . . . . . . . . . . . . . . 17 ⊢ 3 < 4
78	76, 17, 77	ltleii 11297	. . . . . . . . . . . . . . . 16 ⊢ 3 ≤ 4
79		elfz2nn0 13579	. . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 14 ⊢ 3 ∈ 𝑉
82		prelpwi 5407	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ 𝑉 ∧ 3 ∈ 𝑉) → {0, 3} ∈ 𝒫 𝑉)
83		eleq1 2816	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → 𝑝 ∈ 𝒫 𝑉)
86		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = (♯‘{0, 3}))
87		3ne0 12292	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 0
88	87	necomi 2979	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 3
89		3z 12566	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
90		hashprg 14360	. . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . 14 ⊢ (♯‘{0, 3}) = 2
93	86, 92	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑝 = {0, 3} → (♯‘𝑝) = 2)
94	85, 93	jca 511	. . . . . . . . . . . 12 ⊢ (𝑝 = {0, 3} → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
95	74, 94	jaoi 857	. . . . . . . . . . 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 857	. . . . . . . . 9 ⊢ ((𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}) → (𝑝 ∈ 𝒫 𝑉 ∧ (♯‘𝑝) = 2))
98		elun 4116	. . . . . . . . 9 ⊢ (𝑝 ∈ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ↔ (𝑝 ∈ {{0, 1}, {1, 2}} ∨ 𝑝 ∈ {{2, 0}, {0, 3}}))
99		fveqeq2 6867	. . . . . . . . . 10 ⊢ (𝑒 = 𝑝 → ((♯‘𝑒) = 2 ↔ (♯‘𝑝) = 2))
100	99	elrab 3659	. . . . . . . . 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 3950	. . . . . . 7 ⊢ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
103	11, 102	sstrdi 3959	. . . . . 6 ⊢ (ran 𝐸 ⊆ ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) → ran 𝐸 ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})
104	103	anim2i 617	. . . . 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 6515	. . . . 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 6515	. . . . 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 615	. . 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 6755	. . 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 6755	. . 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 847 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∈ wcel 2109 ∃*wmo 2531 ≠ wne 2925 {crab 3405 Vcvv 3447 ∪ cun 3912 ⊆ wss 3914 𝒫 cpw 4563 {cpr 4591 class class class wbr 5107 dom cdm 5638 ran crn 5639 Fn wfn 6506 ⟶wf 6507 –1-1→wf1 6508 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 ≤ cle 11209 ℕcn 12186 2c2 12241 3c3 12242 4c4 12243 ℕ₀cn0 12442 ℤcz 12529 ...cfz 13468 ♯chash 14295 ⟨“cs4 14809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-hash 14296 df-word 14479 df-concat 14536 df-s1 14561 df-s2 14814 df-s3 14815 df-s4 14816

This theorem is referenced by: usgrexmpl 29190

W3C validator