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Theorem lecldbas 20933
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 2621 . . . 4 ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
2 eqid 2621 . . . 4 ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
31, 2leordtval2 20926 . . 3 (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))))
4 fvex 6158 . . . 4 (fi‘ran 𝐹) ∈ V
5 fvex 6158 . . . . . 6 (ordTop‘ ≤ ) ∈ V
6 lecldbas.1 . . . . . . . 8 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ*𝑥))
7 iccf 12214 . . . . . . . . . . 11 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
8 ffn 6002 . . . . . . . . . . 11 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
97, 8ax-mp 5 . . . . . . . . . 10 [,] Fn (ℝ* × ℝ*)
10 ovelrn 6763 . . . . . . . . . 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 3700 . . . . . . . . . . . 12 (𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) = (ℝ* ∖ (𝑎[,]𝑏)))
13 iccordt 20928 . . . . . . . . . . . . 13 (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14 letopuni 20921 . . . . . . . . . . . . . 14 * = (ordTop‘ ≤ )
1514cldopn 20745 . . . . . . . . . . . . 13 ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
1613, 15ax-mp 5 . . . . . . . . . . . 12 (ℝ* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
1712, 16syl6eqel 2706 . . . . . . . . . . 11 (𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
1817rexlimivw 3022 . . . . . . . . . 10 (∃𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
1918rexlimivw 3022 . . . . . . . . 9 (∃𝑎 ∈ ℝ*𝑏 ∈ ℝ* 𝑥 = (𝑎[,]𝑏) → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
2011, 19sylbi 207 . . . . . . . 8 (𝑥 ∈ ran [,] → (ℝ*𝑥) ∈ (ordTop‘ ≤ ))
216, 20fmpti 6339 . . . . . . 7 𝐹:ran [,]⟶(ordTop‘ ≤ )
22 frn 6010 . . . . . . 7 (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
2321, 22ax-mp 5 . . . . . 6 ran 𝐹 ⊆ (ordTop‘ ≤ )
245, 23ssexi 4763 . . . . 5 ran 𝐹 ∈ V
25 eqid 2621 . . . . . . . 8 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞))
26 mnfxr 10040 . . . . . . . . . . 11 -∞ ∈ ℝ*
27 fnovrn 6762 . . . . . . . . . . 11 (([,] Fn (ℝ* × ℝ*) ∧ -∞ ∈ ℝ*𝑦 ∈ ℝ*) → (-∞[,]𝑦) ∈ ran [,])
289, 26, 27mp3an12 1411 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (-∞[,]𝑦) ∈ ran [,])
2926a1i 11 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → -∞ ∈ ℝ*)
30 id 22 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ*𝑦 ∈ ℝ*)
31 pnfxr 10036 . . . . . . . . . . . . . . 15 +∞ ∈ ℝ*
3231a1i 11 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → +∞ ∈ ℝ*)
33 mnfle 11913 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → -∞ ≤ 𝑦)
34 pnfge 11908 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ*𝑦 ≤ +∞)
35 df-icc 12124 . . . . . . . . . . . . . . 15 [,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎𝑐𝑐𝑏)})
36 df-ioc 12122 . . . . . . . . . . . . . . 15 (,] = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎 < 𝑐𝑐𝑏)})
37 xrltnle 10049 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑦 < 𝑧 ↔ ¬ 𝑧𝑦))
38 xrletr 11933 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧𝑦𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39 xrlelttr 11931 . . . . . . . . . . . . . . . 16 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦 < 𝑧) → -∞ < 𝑧))
40 xrltle 11926 . . . . . . . . . . . . . . . . 17 ((-∞ ∈ ℝ*𝑧 ∈ ℝ*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41403adant2 1078 . . . . . . . . . . . . . . . 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 12133 . . . . . . . . . . . . . 14 (((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ 𝑦𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
4429, 30, 32, 33, 34, 43syl32anc 1331 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45 iccmax 12191 . . . . . . . . . . . . 13 (-∞[,]+∞) = ℝ*
4644, 45syl6eq 2671 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ*)
47 iccssxr 12198 . . . . . . . . . . . . 13 (-∞[,]𝑦) ⊆ ℝ*
4835, 36, 37ixxdisj 12132 . . . . . . . . . . . . . 14 ((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
4926, 31, 48mp3an13 1412 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50 uneqdifeq 4029 . . . . . . . . . . . . 13 (((-∞[,]𝑦) ⊆ ℝ* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ* ↔ (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
5147, 49, 50sylancr 694 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ* ↔ (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
5246, 51mpbid 222 . . . . . . . . . . 11 (𝑦 ∈ ℝ* → (ℝ* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
5352eqcomd 2627 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦)))
54 difeq2 3700 . . . . . . . . . . . 12 (𝑥 = (-∞[,]𝑦) → (ℝ*𝑥) = (ℝ* ∖ (-∞[,]𝑦)))
5554eqeq2d 2631 . . . . . . . . . . 11 (𝑥 = (-∞[,]𝑦) → ((𝑦(,]+∞) = (ℝ*𝑥) ↔ (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦))))
5655rspcev 3295 . . . . . . . . . 10 (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
5728, 53, 56syl2anc 692 . . . . . . . . 9 (𝑦 ∈ ℝ* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
58 xrex 11773 . . . . . . . . . . 11 * ∈ V
59 difexg 4768 . . . . . . . . . . 11 (ℝ* ∈ V → (ℝ*𝑥) ∈ V)
6058, 59ax-mp 5 . . . . . . . . . 10 (ℝ*𝑥) ∈ V
616, 60elrnmpti 5336 . . . . . . . . 9 ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ*𝑥))
6257, 61sylibr 224 . . . . . . . 8 (𝑦 ∈ ℝ* → (𝑦(,]+∞) ∈ ran 𝐹)
6325, 62fmpti 6339 . . . . . . 7 (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)):ℝ*⟶ran 𝐹
64 frn 6010 . . . . . . 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 2621 . . . . . . . 8 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))
67 fnovrn 6762 . . . . . . . . . . 11 (([,] Fn (ℝ* × ℝ*) ∧ 𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦[,]+∞) ∈ ran [,])
689, 31, 67mp3an13 1412 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (𝑦[,]+∞) ∈ ran [,])
69 df-ico 12123 . . . . . . . . . . . . . . 15 [,) = (𝑎 ∈ ℝ*, 𝑏 ∈ ℝ* ↦ {𝑐 ∈ ℝ* ∣ (𝑎𝑐𝑐 < 𝑏)})
70 xrlenlt 10047 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → (𝑦𝑧 ↔ ¬ 𝑧 < 𝑦))
71 xrltletr 11932 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧 < 𝑦𝑦 ≤ +∞) → 𝑧 < +∞))
72 xrltle 11926 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
73723adant2 1078 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
7471, 73syld 47 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑧 < 𝑦𝑦 ≤ +∞) → 𝑧 ≤ +∞))
75 xrletr 11933 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ*𝑧 ∈ ℝ*) → ((-∞ ≤ 𝑦𝑦𝑧) → -∞ ≤ 𝑧))
7669, 35, 70, 35, 74, 75ixxun 12133 . . . . . . . . . . . . . 14 (((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ ≤ 𝑦𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
7729, 30, 32, 33, 34, 76syl32anc 1331 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
78 uncom 3735 . . . . . . . . . . . . 13 ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
7977, 78, 453eqtr3g 2678 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ*)
80 iccssxr 12198 . . . . . . . . . . . . 13 (𝑦[,]+∞) ⊆ ℝ*
81 incom 3783 . . . . . . . . . . . . . 14 ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
8269, 35, 70ixxdisj 12132 . . . . . . . . . . . . . . 15 ((-∞ ∈ ℝ*𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
8326, 31, 82mp3an13 1412 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
8481, 83syl5eq 2667 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
85 uneqdifeq 4029 . . . . . . . . . . . . 13 (((𝑦[,]+∞) ⊆ ℝ* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ* ↔ (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
8680, 84, 85sylancr 694 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ* ↔ (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
8779, 86mpbid 222 . . . . . . . . . . 11 (𝑦 ∈ ℝ* → (ℝ* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
8887eqcomd 2627 . . . . . . . . . 10 (𝑦 ∈ ℝ* → (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞)))
89 difeq2 3700 . . . . . . . . . . . 12 (𝑥 = (𝑦[,]+∞) → (ℝ*𝑥) = (ℝ* ∖ (𝑦[,]+∞)))
9089eqeq2d 2631 . . . . . . . . . . 11 (𝑥 = (𝑦[,]+∞) → ((-∞[,)𝑦) = (ℝ*𝑥) ↔ (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞))))
9190rspcev 3295 . . . . . . . . . 10 (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
9268, 88, 91syl2anc 692 . . . . . . . . 9 (𝑦 ∈ ℝ* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
936, 60elrnmpti 5336 . . . . . . . . 9 ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ*𝑥))
9492, 93sylibr 224 . . . . . . . 8 (𝑦 ∈ ℝ* → (-∞[,)𝑦) ∈ ran 𝐹)
9566, 94fmpti 6339 . . . . . . 7 (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)):ℝ*⟶ran 𝐹
96 frn 6010 . . . . . . 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 3766 . . . . 5 (ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
99 fiss 8274 . . . . 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 707 . . . 4 (fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)
101 tgss 20683 . . . 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 707 . . 3 (topGen‘(fi‘(ran (𝑦 ∈ ℝ* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹))
1033, 102eqsstri 3614 . 2 (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
104 letop 20920 . . 3 (ordTop‘ ≤ ) ∈ Top
105 tgfiss 20706 . . 3 (((ordTop‘ ≤ ) ∈ Top ∧ ran 𝐹 ⊆ (ordTop‘ ≤ )) → (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ ))
106104, 23, 105mp2an 707 . 2 (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ )
107103, 106eqssi 3599 1 (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673   × cxp 5072  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  ficfi 8260  +∞cpnf 10015  -∞cmnf 10016  *cxr 10017   < clt 10018  cle 10019  (,]cioc 12118  [,)cico 12119  [,]cicc 12120  topGenctg 16019  ordTopcordt 16080  Topctop 20617  Clsdccld 20730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-ioc 12122  df-ico 12123  df-icc 12124  df-topgen 16025  df-ordt 16082  df-ps 17121  df-tsr 17122  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733
This theorem is referenced by: (None)
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