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Theorem tgdom 20071
 Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
tgdom

Proof of Theorem tgdom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4585 . 2
2 inss1 3643 . . . . 5
3 vex 3034 . . . . . . . 8
43pwex 4584 . . . . . . 7
54inex2 4538 . . . . . 6
65elpw 3948 . . . . 5
72, 6mpbir 214 . . . 4
87a1i 11 . . 3
9 unieq 4198 . . . . . . 7
109adantl 473 . . . . . 6
11 eltg4i 20052 . . . . . . 7
1211ad2antrr 740 . . . . . 6
13 eltg4i 20052 . . . . . . 7
1413ad2antlr 741 . . . . . 6
1510, 12, 143eqtr4d 2515 . . . . 5
1615ex 441 . . . 4
17 pweq 3945 . . . . 5
1817ineq2d 3625 . . . 4
1916, 18impbid1 208 . . 3
208, 19dom2 7630 . 2
211, 20syl 17 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  cvv 3031   cin 3389   wss 3390  cpw 3942  cuni 4190   class class class wbr 4395  cfv 5589   cdom 7585  ctg 15414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-dom 7589  df-topgen 15420 This theorem is referenced by:  2ndcredom  20542  kelac2lem  35993
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