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Theorem dis2ndc 22827
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 8906 . 2 (𝑋 β‰Ό Ο‰ β†’ 𝑋 ∈ V)
2 pwexr 7700 . 2 (𝒫 𝑋 ∈ 2ndΟ‰ β†’ 𝑋 ∈ V)
3 vsnex 5387 . . . . . . . 8 {π‘₯} ∈ V
432a1i 12 . . . . . . 7 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 β†’ {π‘₯} ∈ V))
5 vex 3448 . . . . . . . . . 10 π‘₯ ∈ V
65sneqr 4799 . . . . . . . . 9 ({π‘₯} = {𝑦} β†’ π‘₯ = 𝑦)
7 sneq 4597 . . . . . . . . 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 8935 . . . . . 6 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):𝑋–1-1β†’V)
11 f1f1orn 6796 . . . . . 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 8914 . . . . 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 9066 . . . 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 22361 . . . . . . 7 (𝑋 ∈ V β†’ 𝒫 𝑋 ∈ Top)
18 simpr 486 . . . . . . . . . 10 ((𝑋 ∈ V ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
195snelpw 5403 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↔ {π‘₯} ∈ 𝒫 𝑋)
2018, 19sylib 217 . . . . . . . . 9 ((𝑋 ∈ V ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝒫 𝑋)
2120fmpttd 7064 . . . . . . . 8 (𝑋 ∈ V β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):π‘‹βŸΆπ’« 𝑋)
2221frnd 6677 . . . . . . 7 (𝑋 ∈ V β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) βŠ† 𝒫 𝑋)
23 elpwi 4568 . . . . . . . . . . . . 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 3946 . . . . . . . . . . 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 4597 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ {π‘₯} = {𝑧})
2928rspceeqv 3596 . . . . . . . . . . 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 5387 . . . . . . . . . . 11 {𝑧} ∈ V
32 eqid 2733 . . . . . . . . . . . 12 (π‘₯ ∈ 𝑋 ↦ {π‘₯}) = (π‘₯ ∈ 𝑋 ↦ {π‘₯})
3332elrnmpt 5912 . . . . . . . . . . 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 4624 . . . . . . . . . 10 𝑧 ∈ {𝑧}
3736a1i 11 . . . . . . . . 9 ((𝑋 ∈ V ∧ (𝑦 ∈ 𝒫 𝑋 ∧ 𝑧 ∈ 𝑦)) β†’ 𝑧 ∈ {𝑧})
3825snssd 4770 . . . . . . . . 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 3970 . . . . . . . . . . 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 3580 . . . . . . . . 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 3194 . . . . . . 7 (𝑋 ∈ V β†’ βˆ€π‘¦ ∈ 𝒫 π‘‹βˆ€π‘§ ∈ 𝑦 βˆƒπ‘€ ∈ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯})(𝑧 ∈ 𝑀 ∧ 𝑀 βŠ† 𝑦))
45 basgen2 22355 . . . . . . 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 22336 . . . . . . 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 22815 . . . . . 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 22813 . . . . . 6 (𝒫 𝑋 ∈ 2ndΟ‰ ↔ βˆƒπ‘ ∈ TopBases (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋))
55 vex 3448 . . . . . . . . 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 4624 . . . . . . . . . . . . 13 π‘₯ ∈ {π‘₯}
61 tg2 22331 . . . . . . . . . . . . 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 4770 . . . . . . . . . . . . . 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 3962 . . . . . . . . . . . . 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 3155 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) ∧ π‘₯ ∈ 𝑋) β†’ {π‘₯} ∈ 𝑏)
7069fmpttd 7064 . . . . . . . . . 10 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ (π‘₯ ∈ 𝑋 ↦ {π‘₯}):π‘‹βŸΆπ‘)
7170frnd 6677 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑏 ∈ TopBases) ∧ (𝑏 β‰Ό Ο‰ ∧ (topGenβ€˜π‘) = 𝒫 𝑋)) β†’ ran (π‘₯ ∈ 𝑋 ↦ {π‘₯}) βŠ† 𝑏)
72 ssdomg 8943 . . . . . . . . 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 8950 . . . . . . . 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 3151 . . . . . 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 3061  βˆƒwrex 3070  Vcvv 3444   βŠ† wss 3911  π’« cpw 4561  {csn 4587   class class class wbr 5106   ↦ cmpt 5189  ran crn 5635  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  Ο‰com 7803   β‰ˆ cen 8883   β‰Ό cdom 8884  topGenctg 17324  Topctop 22258  TopBasesctb 22311  2ndΟ‰c2ndc 22805
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-dom 8888  df-topgen 17330  df-top 22259  df-bases 22312  df-2ndc 22807
This theorem is referenced by: (None)
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