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Theorem ontgsucval 26174
Description: The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
Assertion
Ref Expression
ontgsucval  |-  ( A  e.  On  ->  ( topGen `
 suc  A )  =  suc  A )

Proof of Theorem ontgsucval
StepHypRef Expression
1 suceloni 4785 . . 3  |-  ( A  e.  On  ->  suc  A  e.  On )
2 ontgval 26173 . . 3  |-  ( suc 
A  e.  On  ->  (
topGen `  suc  A )  =  suc  U. suc  A )
31, 2syl 16 . 2  |-  ( A  e.  On  ->  ( topGen `
 suc  A )  =  suc  U. suc  A
)
4 eloni 4583 . . . 4  |-  ( A  e.  On  ->  Ord  A )
5 ordunisuc 4804 . . . 4  |-  ( Ord 
A  ->  U. suc  A  =  A )
64, 5syl 16 . . 3  |-  ( A  e.  On  ->  U. suc  A  =  A )
7 suceq 4638 . . 3  |-  ( U. suc  A  =  A  ->  suc  U. suc  A  =  suc  A )
86, 7syl 16 . 2  |-  ( A  e.  On  ->  suc  U.
suc  A  =  suc  A )
93, 8eqtrd 2467 1  |-  ( A  e.  On  ->  ( topGen `
 suc  A )  =  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   U.cuni 4007   Ord word 4572   Oncon0 4573   suc csuc 4575   ` cfv 5446   topGenctg 13657
This theorem is referenced by:  onsuctop  26175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-topgen 13659
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