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Theorem hmphi 17766
Description: If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphi  |-  ( F  e.  ( J  Homeo  K )  ->  J  ~=  K )

Proof of Theorem hmphi
StepHypRef Expression
1 ne0i 3598 . 2  |-  ( F  e.  ( J  Homeo  K )  ->  ( J  Homeo  K )  =/=  (/) )
2 hmph 17765 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
31, 2sylibr 204 1  |-  ( F  e.  ( J  Homeo  K )  ->  J  ~=  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    =/= wne 2571   (/)c0 3592   class class class wbr 4176  (class class class)co 6044    Homeo chmeo 17742    ~= chmph 17743
This theorem is referenced by:  hmphref  17770  hmphsym  17771  hmphtr  17772  indishmph  17787  ptcmpfi  17802  t0kq  17807  kqhmph  17808  xrhmph  18929  xrge0hmph  24275  reheibor  26442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-1o 6687  df-hmeo 17744  df-hmph 17745
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