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Theorem dis2ndc 22964
Description: A discrete space is second-countable iff it is countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
Assertion
Ref Expression
dis2ndc (𝑋 β‰Ό Ο‰ ↔ 𝒫 𝑋 ∈ 2ndΟ‰)
Proof of Theorem dis2ndc
Dummy variables 𝑀 𝑏 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctex 8959 . 2 (𝑋 β‰Ό Ο‰ β†’ 𝑋 ∈ V)
2 pwexr 7752 . 2 (𝒫 𝑋 ∈ 2ndΟ‰ β†’ 𝑋 ∈ V)
3 vsnex 5430 . . . . . . . 8 {π‘₯} ∈ V
432a1i 12 . . . . . . 7 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 β†’ {π‘₯} ∈ V))
5 vex 3479 . . . . . . . . . 10 π‘₯ ∈ V
65sneqr 4842 . . . . . . . . 9 ({π‘₯} = {𝑦} β†’ π‘₯ = 𝑦)
7 sneq 4639 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ {π‘₯} = {𝑦})
86, 7impbii 208 . . . . . . . 8 ({π‘₯} = {𝑦} ↔ π‘₯ = 𝑦)
982a1i 12 . . . . . . 7 (𝑋 ∈ V β†’ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ ({π‘₯} = {𝑦} ↔ π‘₯ = 𝑦)))
104, 9dom2lem 8988 . . . . . 6 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):𝑋–1-1β†’V)
11 f1f1orn 6845 . . . . . 6 ((π‘₯ ∈ 𝑋 ↦ {π‘₯}):𝑋–1-1β†’V β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):𝑋–1-1-ontoβ†’ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}))
1210, 11syl 17 . . . . 5 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):𝑋–1-1-ontoβ†’ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}))
13 f1oeng 8967 . . . . 5 ((𝑋 ∈ V ∧ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):𝑋–1-1-ontoβ†’ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) β†’ 𝑋 β‰ˆ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}))
1412, 13mpdan 686 . . . 4 (𝑋 ∈ V β†’ 𝑋 β‰ˆ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}))
15 domen1 9119 . . . 4 (𝑋 β‰ˆ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β†’ (𝑋 β‰Ό Ο‰ ↔ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰))
1614, 15syl 17 . . 3 (𝑋 ∈ V β†’ (𝑋 β‰Ό Ο‰ ↔ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰))
17 distop 22498 . . . . . . 7 (𝑋 ∈ V β†’ 𝒫 𝑋 ∈ Top)
18 simpr 486 . . . . . . . . . 10 ((𝑋 ∈ V ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
195snelpw 5446 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↔ {π‘₯} ∈ 𝒫 𝑋)
2018, 19sylib 217 . . . . . . . . 9 ((𝑋 ∈ V ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝒫 𝑋)
2120fmpttd 7115 . . . . . . . 8 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):π‘‹βŸΆπ’« 𝑋)
2221frnd 6726 . . . . . . 7 (𝑋 ∈ V β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) βŠ† 𝒫 𝑋)
23 elpwi 4610 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 𝑋 β†’ 𝑦 βŠ† 𝑋)
2423ad2antrl 727 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ 𝑦 βŠ† 𝑋)
25 simprr 772 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ 𝑧 ∈ 𝑦)
2624, 25sseldd 3984 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ 𝑧 ∈ 𝑋)
27 eqidd 2734 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ {𝑧} = {𝑧})
28 sneq 4639 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ {π‘₯} = {𝑧})
2928rspceeqv 3634 . . . . . . . . . . 11 ((𝑧 ∈ 𝑋 ∧ {𝑧} = {𝑧}) β†’ βˆƒπ‘₯ ∈ 𝑋 {𝑧} = {π‘₯})
3026, 27, 29syl2anc 585 . . . . . . . . . 10 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ βˆƒπ‘₯ ∈ 𝑋 {𝑧} = {π‘₯})
31 vsnex 5430 . . . . . . . . . . 11 {𝑧} ∈ V
32 eqid 2733 . . . . . . . . . . . 12 (π‘₯ ∈ 𝑋 ↦ {π‘₯}) = (π‘₯ ∈ 𝑋 ↦ {π‘₯})
3332elrnmpt 5956 . . . . . . . . . . 11 ({𝑧} ∈ V β†’ ({𝑧} ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) ↔ βˆƒπ‘₯ ∈ 𝑋 {𝑧} = {π‘₯}))
3431, 33ax-mp 5 . . . . . . . . . 10 ({𝑧} ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) ↔ βˆƒπ‘₯ ∈ 𝑋 {𝑧} = {π‘₯})
3530, 34sylibr 233 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ {𝑧} ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}))
36 vsnid 4666 . . . . . . . . . 10 𝑧 ∈ {𝑧}
3736a1i 11 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ 𝑧 ∈ {𝑧})
3825snssd 4813 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ {𝑧} βŠ† 𝑦)
39 eleq2 2823 . . . . . . . . . . 11 (𝑀 = {𝑧} β†’ (𝑧 ∈ 𝑀 ↔ 𝑧 ∈ {𝑧}))
40 sseq1 4008 . . . . . . . . . . 11 (𝑀 = {𝑧} β†’ (𝑀 βŠ† 𝑦 ↔ {𝑧} βŠ† 𝑦))
4139, 40anbi12d 632 . . . . . . . . . 10 (𝑀 = {𝑧} β†’ ((𝑧 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑦) ↔ (𝑧 ∈ {𝑧} ∧ {𝑧} βŠ† 𝑦)))
4241rspcev 3613 . . . . . . . . 9 (({𝑧} ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) ∧ (𝑧 ∈ {𝑧} ∧ {𝑧} βŠ† 𝑦)) β†’ βˆƒπ‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})(𝑧 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑦))
4335, 37, 38, 42syl12anc 836 . . . . . . . 8 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ βˆƒπ‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})(𝑧 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑦))
4443ralrimivva 3201 . . . . . . 7 (𝑋 ∈ V β†’ βˆ€π‘¦ ∈ 𝒫 π‘‹βˆ€π‘§ ∈ 𝑦 βˆƒπ‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})(𝑧 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑦))
45 basgen2 22492 . . . . . . 7 ((𝒫 𝑋 ∈ Top ∧ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) βŠ† 𝒫 𝑋 ∧ βˆ€π‘¦ ∈ 𝒫 π‘‹βˆ€π‘§ ∈ 𝑦 βˆƒπ‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})(𝑧 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑦)) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) = 𝒫 𝑋)
4617, 22, 44, 45syl3anc 1372 . . . . . 6 (𝑋 ∈ V β†’ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) = 𝒫 𝑋)
4746adantr 482 . . . . 5 ((𝑋 ∈ V ∧ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) = 𝒫 𝑋)
4846, 17eqeltrd 2834 . . . . . . 7 (𝑋 ∈ V β†’ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) ∈ Top)
49 tgclb 22473 . . . . . . 7 (ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) ∈ TopBases ↔ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) ∈ Top)
5048, 49sylibr 233 . . . . . 6 (𝑋 ∈ V β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) ∈ TopBases)
51 2ndci 22952 . . . . . 6 ((ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) ∈ TopBases ∧ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) ∈ 2ndΟ‰)
5250, 51sylan 581 . . . . 5 ((𝑋 ∈ V ∧ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})) ∈ 2ndΟ‰)
5347, 52eqeltrrd 2835 . . . 4 ((𝑋 ∈ V ∧ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰) β†’ 𝒫 𝑋 ∈ 2ndΟ‰)
54 is2ndc 22950 . . . . . 6 (𝒫 𝑋 ∈ 2ndΟ‰ ↔ βˆƒπ‘ ∈ TopBases (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋))
55 vex 3479 . . . . . . . . 9 𝑏 ∈ V
56 simpr 486 . . . . . . . . . . . . . . 15 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
5756, 19sylib 217 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝒫 𝑋)
58 simplrr 777 . . . . . . . . . . . . . 14 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ (topGenβ€˜π‘) = 𝒫 𝑋)
5957, 58eleqtrrd 2837 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ (topGenβ€˜π‘))
60 vsnid 4666 . . . . . . . . . . . . 13 π‘₯ ∈ {π‘₯}
61 tg2 22468 . . . . . . . . . . . . 13 (({π‘₯} ∈ (topGenβ€˜π‘) ∧ π‘₯ ∈ {π‘₯}) β†’ βˆƒπ‘¦ ∈ 𝑏 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))
6259, 60, 61sylancl 587 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑏 (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))
63 simprrl 780 . . . . . . . . . . . . . . 15 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))) β†’ π‘₯ ∈ 𝑦)
6463snssd 4813 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))) β†’ {π‘₯} βŠ† 𝑦)
65 simprrr 781 . . . . . . . . . . . . . 14 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))) β†’ 𝑦 βŠ† {π‘₯})
6664, 65eqssd 4000 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))) β†’ {π‘₯} = 𝑦)
67 simprl 770 . . . . . . . . . . . . 13 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))) β†’ 𝑦 ∈ 𝑏)
6866, 67eqeltrd 2834 . . . . . . . . . . . 12 (((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝑏 ∧ (π‘₯ ∈ 𝑦 ∧ 𝑦 βŠ† {π‘₯}))) β†’ {π‘₯} ∈ 𝑏)
6962, 68rexlimddv 3162 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝑏)
7069fmpttd 7115 . . . . . . . . . 10 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):π‘‹βŸΆπ‘)
7170frnd 6726 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) βŠ† 𝑏)
72 ssdomg 8996 . . . . . . . . 9 (𝑏 ∈ V β†’ (ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) βŠ† 𝑏 β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό 𝑏))
7355, 71, 72mpsyl 68 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό 𝑏)
74 simprl 770 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ 𝑏 β‰Ό Ο‰)
75 domtr 9003 . . . . . . . 8 ((ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό 𝑏 ∧ 𝑏 β‰Ό Ο‰) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰)
7673, 74, 75syl2anc 585 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰)
7776rexlimdva2 3158 . . . . . 6 (𝑋 ∈ V β†’ (βˆƒπ‘ ∈ TopBases (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰))
7854, 77biimtrid 241 . . . . 5 (𝑋 ∈ V β†’ (𝒫 𝑋 ∈ 2ndΟ‰ β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰))
7978imp 408 . . . 4 ((𝑋 ∈ V ∧ 𝒫 𝑋 ∈ 2ndΟ‰) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰)
8053, 79impbida 800 . . 3 (𝑋 ∈ V β†’ (ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) β‰Ό Ο‰ ↔ 𝒫 𝑋 ∈ 2ndΟ‰))
8116, 80bitrd 279 . 2 (𝑋 ∈ V β†’ (𝑋 β‰Ό Ο‰ ↔ 𝒫 𝑋 ∈ 2ndΟ‰))
821, 2, 81pm5.21nii 380 1 (𝑋 β‰Ό Ο‰ ↔ 𝒫 𝑋 ∈ 2ndΟ‰)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  {csn 4629   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678  β€“1-1β†’wf1 6541  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  Ο‰com 7855   β‰ˆ cen 8936   β‰Ό cdom 8937  topGenctg 17383  Topctop 22395  TopBasesctb 22448  2ndΟ‰c2ndc 22942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-topgen 17389  df-top 22396  df-bases 22449  df-2ndc 22944
This theorem is referenced by: (None)
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