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Theorem lecldbas 21244
Description: The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
lecldbas.1 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ*𝑥))
Assertion
Ref Expression
lecldbas (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Proof of Theorem lecldbas
Dummy variables 𝑎 𝑏 𝑐 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2771 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
31, 2leordtval2 21237 . . 3 (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))))
4 fvex 6342 . . . 4 (fi‘ran 𝐹) ∈ V
5 fvex 6342 . . . . . 6 (ordTop‘ ≤ ) ∈ V
6 lecldbas.1 . . . . . . . 8 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ*𝑥))
7 iccf 12478 . . . . . . . . . . 11 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
8 ffn 6185 . . . . . . . . . . 11 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
97, 8ax-mp 5 . . . . . . . . . 10 [,] Fn (ℝ* × ℝ*)
10 ovelrn 6957 . . . . . . . . . 10 ([,] Fn (ℝ* × ℝ*) → (𝑥 ∈ ran [,] ↔ ∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏)))
119, 10ax-mp 5 . . . . . . . . 9 (𝑥 ∈ ran [,] ↔ ∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏))
12 difeq2 3873 . . . . . . . . . . . 12 (𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) = (ℝ* ∖ (𝑎[,]𝑏)))
13 iccordt 21239 . . . . . . . . . . . . 13 (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14 letopuni 21232 . . . . . . . . . . . . . 14 * = (ordTop‘ ≤ )
1514cldopn 21056 . . . . . . . . . . . . 13 ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
1613, 15ax-mp 5 . . . . . . . . . . . 12 (ℝ* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
1712, 16syl6eqel 2858 . . . . . . . . . . 11 (𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
1817rexlimivw 3177 . . . . . . . . . 10 (∃𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
1918rexlimivw 3177 . . . . . . . . 9 (∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
2011, 19sylbi 207 . . . . . . . 8 (𝑥 ∈ ran [,] → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
216, 20fmpti 6525 . . . . . . 7 𝐹:ran [,]⟶(ordTop‘ ≤ )
22 frn 6193 . . . . . . 7 (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
2321, 22ax-mp 5 . . . . . 6 ran 𝐹 ⊆ (ordTop‘ ≤ )
245, 23ssexi 4937 . . . . 5 ran 𝐹 ∈ V
25 eqid 2771 . . . . . . . 8 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
26 mnfxr 10298 . . . . . . . . . . 11 -∞ ∈ ℝ*
27 fnovrn 6956 . . . . . . . . . . 11 (([,] Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ*𝑦 ∈ ℝ*) → (-∞[,]𝑦) ∈ ran [,])
289, 26, 27mp3an12 1562 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (-∞[,]𝑦) ∈ ran [,])
2926a1i 11 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → -∞ ∈ ℝ*)
30 id 22 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ*𝑦 ∈ ℝ*)
31 pnfxr 10294 . . . . . . . . . . . . . . 15 +∞ ∈ ℝ*
3231a1i 11 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → +∞ ∈ ℝ*)
33 mnfle 12174 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → -∞ ≤ 𝑦)
34 pnfge 12169 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ*𝑦 ≤ +∞)
35 df-icc 12387 . . . . . . . . . . . . . . 15 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎𝑐𝑐𝑏)})
36 df-ioc 12385 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
37 xrltnle 10307 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑦 < 𝑧 ↔ ¬ 𝑧𝑦))
38 xrletr 12194 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧𝑦𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39 xrlelttr 12192 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦 < 𝑧) → -∞ < 𝑧))
40 xrltle 12187 . . . . . . . . . . . . . . . . 17 ((-∞ ∈ ℝ*𝑧 ∈ ℝ*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41403adant2 1125 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
4239, 41syld 47 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦 < 𝑧) → -∞ ≤ 𝑧))
4335, 36, 37, 35, 38, 42ixxun 12396 . . . . . . . . . . . . . 14 (((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ 𝑦𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
4429, 30, 32, 33, 34, 43syl32anc 1484 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45 iccmax 12454 . . . . . . . . . . . . 13 (-∞[,]+∞) = ℝ*
4644, 45syl6eq 2821 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ*)
47 iccssxr 12461 . . . . . . . . . . . . 13 (-∞[,]𝑦) ⊆ ℝ*
4835, 36, 37ixxdisj 12395 . . . . . . . . . . . . . 14 ((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
4926, 31, 48mp3an13 1563 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50 uneqdifeq 4199 . . . . . . . . . . . . 13 (((-∞[,]𝑦) ⊆ ℝ* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ* ↔ (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
5147, 49, 50sylancr 575 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ* ↔ (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
5246, 51mpbid 222 . . . . . . . . . . 11 (𝑦 ∈ ℝ* → (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
5352eqcomd 2777 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦)))
54 difeq2 3873 . . . . . . . . . . . 12 (𝑥 = (-∞[,]𝑦) → (ℝ*𝑥) = (ℝ* ∖ (-∞[,]𝑦)))
5554eqeq2d 2781 . . . . . . . . . . 11 (𝑥 = (-∞[,]𝑦) → ((𝑦(,]+∞) = (ℝ*𝑥) ↔ (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦))))
5655rspcev 3460 . . . . . . . . . 10 (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
5728, 53, 56syl2anc 573 . . . . . . . . 9 (𝑦 ∈ ℝ* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
58 xrex 12032 . . . . . . . . . . 11 * ∈ V
59 difexg 4942 . . . . . . . . . . 11 (ℝ* ∈ V → (ℝ*𝑥) ∈ V)
6058, 59ax-mp 5 . . . . . . . . . 10 (ℝ*𝑥) ∈ V
616, 60elrnmpti 5514 . . . . . . . . 9 ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
6257, 61sylibr 224 . . . . . . . 8 (𝑦 ∈ ℝ* → (𝑦(,]+∞) ∈ ran 𝐹)
6325, 62fmpti 6525 . . . . . . 7 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)):ℝ*⟶ran 𝐹
64 frn 6193 . . . . . . 7 ((𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)):ℝ*⟶ran 𝐹 → ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ⊆ ran 𝐹)
6563, 64ax-mp 5 . . . . . 6 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ⊆ ran 𝐹
66 eqid 2771 . . . . . . . 8 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
67 fnovrn 6956 . . . . . . . . . . 11 (([,] Fn (ℝ* × ℝ*) ∧ 𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦[,]+∞) ∈ ran [,])
689, 31, 67mp3an13 1563 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (𝑦[,]+∞) ∈ ran [,])
69 df-ico 12386 . . . . . . . . . . . . . . 15 [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎𝑐𝑐 < 𝑏)})
70 xrlenlt 10305 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑦𝑧 ↔ ¬ 𝑧 < 𝑦))
71 xrltletr 12193 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧 < 𝑦𝑦 ≤ +∞) → 𝑧 < +∞))
72 xrltle 12187 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
73723adant2 1125 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
7471, 73syld 47 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧 < 𝑦𝑦 ≤ +∞) → 𝑧 ≤ +∞))
75 xrletr 12194 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦𝑧) → -∞ ≤ 𝑧))
7669, 35, 70, 35, 74, 75ixxun 12396 . . . . . . . . . . . . . 14 (((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ 𝑦𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
7729, 30, 32, 33, 34, 76syl32anc 1484 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
78 uncom 3908 . . . . . . . . . . . . 13 ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
7977, 78, 453eqtr3g 2828 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ*)
80 iccssxr 12461 . . . . . . . . . . . . 13 (𝑦[,]+∞) ⊆ ℝ*
81 incom 3956 . . . . . . . . . . . . . 14 ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
8269, 35, 70ixxdisj 12395 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
8326, 31, 82mp3an13 1563 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
8481, 83syl5eq 2817 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
85 uneqdifeq 4199 . . . . . . . . . . . . 13 (((𝑦[,]+∞) ⊆ ℝ* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ* ↔ (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
8680, 84, 85sylancr 575 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ* ↔ (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
8779, 86mpbid 222 . . . . . . . . . . 11 (𝑦 ∈ ℝ* → (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
8887eqcomd 2777 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞)))
89 difeq2 3873 . . . . . . . . . . . 12 (𝑥 = (𝑦[,]+∞) → (ℝ*𝑥) = (ℝ* ∖ (𝑦[,]+∞)))
9089eqeq2d 2781 . . . . . . . . . . 11 (𝑥 = (𝑦[,]+∞) → ((-∞[,)𝑦) = (ℝ*𝑥) ↔ (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞))))
9190rspcev 3460 . . . . . . . . . 10 (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
9268, 88, 91syl2anc 573 . . . . . . . . 9 (𝑦 ∈ ℝ* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
936, 60elrnmpti 5514 . . . . . . . . 9 ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
9492, 93sylibr 224 . . . . . . . 8 (𝑦 ∈ ℝ* → (-∞[,)𝑦) ∈ ran 𝐹)
9566, 94fmpti 6525 . . . . . . 7 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)):ℝ*⟶ran 𝐹
96 frn 6193 . . . . . . 7 ((𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)):ℝ*⟶ran 𝐹 → ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ⊆ ran 𝐹)
9795, 96ax-mp 5 . . . . . 6 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) ⊆ ran 𝐹
9865, 97unssi 3939 . . . . 5 (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
99 fiss 8486 . . . . 5 ((ran 𝐹 ∈ V ∧ (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹) → (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹))
10024, 98, 99mp2an 672 . . . 4 (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)
101 tgss 20993 . . . 4 (((fi‘ran 𝐹) ∈ V ∧ (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)) → (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹)))
1024, 100, 101mp2an 672 . . 3 (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹))
1033, 102eqsstri 3784 . 2 (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
104 letop 21231 . . 3 (ordTop‘ ≤ ) ∈ Top
105 tgfiss 21016 . . 3 (((ordTop‘ ≤ ) ∈ Top ∧ ran 𝐹 ⊆ (ordTop‘ ≤ )) → (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ ))
106104, 23, 105mp2an 672 . 2 (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ )
107103, 106eqssi 3768 1 (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863   × cxp 5247  ran crn 5250   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  ficfi 8472  +∞cpnf 10273  -∞cmnf 10274  *cxr 10275   < clt 10276  cle 10277  (,]cioc 12381  [,)cico 12382  [,]cicc 12383  topGenctg 16306  ordTopcordt 16367  Topctop 20918  Clsdccld 21041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8473  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-ioc 12385  df-ico 12386  df-icc 12387  df-topgen 16312  df-ordt 16369  df-ps 17408  df-tsr 17409  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044
This theorem is referenced by: (None)
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