![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2ndcredom | Structured version Visualization version GIF version |
Description: A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
2ndcredom | β’ (π½ β 2ndΟ β π½ βΌ β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | is2ndc 22813 | . 2 β’ (π½ β 2ndΟ β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) | |
2 | tgdom 22344 | . . . . . 6 β’ (π₯ β TopBases β (topGenβπ₯) βΌ π« π₯) | |
3 | simpr 486 | . . . . . . . . 9 β’ ((π₯ β TopBases β§ π₯ βΌ Ο) β π₯ βΌ Ο) | |
4 | nnenom 13891 | . . . . . . . . . 10 β’ β β Ο | |
5 | 4 | ensymi 8947 | . . . . . . . . 9 β’ Ο β β |
6 | domentr 8956 | . . . . . . . . 9 β’ ((π₯ βΌ Ο β§ Ο β β) β π₯ βΌ β) | |
7 | 3, 5, 6 | sylancl 587 | . . . . . . . 8 β’ ((π₯ β TopBases β§ π₯ βΌ Ο) β π₯ βΌ β) |
8 | pwdom 9076 | . . . . . . . 8 β’ (π₯ βΌ β β π« π₯ βΌ π« β) | |
9 | 7, 8 | syl 17 | . . . . . . 7 β’ ((π₯ β TopBases β§ π₯ βΌ Ο) β π« π₯ βΌ π« β) |
10 | rpnnen 16114 | . . . . . . . 8 β’ β β π« β | |
11 | 10 | ensymi 8947 | . . . . . . 7 β’ π« β β β |
12 | domentr 8956 | . . . . . . 7 β’ ((π« π₯ βΌ π« β β§ π« β β β) β π« π₯ βΌ β) | |
13 | 9, 11, 12 | sylancl 587 | . . . . . 6 β’ ((π₯ β TopBases β§ π₯ βΌ Ο) β π« π₯ βΌ β) |
14 | domtr 8950 | . . . . . 6 β’ (((topGenβπ₯) βΌ π« π₯ β§ π« π₯ βΌ β) β (topGenβπ₯) βΌ β) | |
15 | 2, 13, 14 | syl2an2r 684 | . . . . 5 β’ ((π₯ β TopBases β§ π₯ βΌ Ο) β (topGenβπ₯) βΌ β) |
16 | breq1 5109 | . . . . 5 β’ ((topGenβπ₯) = π½ β ((topGenβπ₯) βΌ β β π½ βΌ β)) | |
17 | 15, 16 | syl5ibcom 244 | . . . 4 β’ ((π₯ β TopBases β§ π₯ βΌ Ο) β ((topGenβπ₯) = π½ β π½ βΌ β)) |
18 | 17 | expimpd 455 | . . 3 β’ (π₯ β TopBases β ((π₯ βΌ Ο β§ (topGenβπ₯) = π½) β π½ βΌ β)) |
19 | 18 | rexlimiv 3142 | . 2 β’ (βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½) β π½ βΌ β) |
20 | 1, 19 | sylbi 216 | 1 β’ (π½ β 2ndΟ β π½ βΌ β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3070 π« cpw 4561 class class class wbr 5106 βcfv 6497 Οcom 7803 β cen 8883 βΌ cdom 8884 βcr 11055 βcn 12158 topGenctg 17324 TopBasesctb 22311 2ndΟc2ndc 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 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 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-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 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-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 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-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-omul 8418 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-acn 9883 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-topgen 17330 df-2ndc 22807 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |