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Theorem dissnref 20555
 Description: The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)
Hypothesis
Ref Expression
dissnref.c
Assertion
Ref Expression
dissnref
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem dissnref
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpr 463 . . 3
2 dissnref.c . . . 4
32unisngl 20554 . . 3
41, 3syl6eq 2503 . 2
5 simplr 763 . . . . . 6
6 simprr 767 . . . . . . 7
76snssd 4120 . . . . . 6
85, 7eqsstrd 3468 . . . . 5
9 simplr 763 . . . . . . 7
10 simp-4r 778 . . . . . . 7
119, 10eleqtrrd 2534 . . . . . 6
12 eluni2 4205 . . . . . 6
1311, 12sylib 200 . . . . 5
148, 13reximddv 2865 . . . 4
152abeq2i 2565 . . . . . 6
1615biimpi 198 . . . . 5
1716adantl 468 . . . 4
1814, 17r19.29a 2934 . . 3
1918ralrimiva 2804 . 2
20 pwexg 4590 . . . . 5
21 simpr 463 . . . . . . . . 9
22 snelpwi 4648 . . . . . . . . . 10
2322ad2antlr 734 . . . . . . . . 9
2421, 23eqeltrd 2531 . . . . . . . 8
2524, 16r19.29a 2934 . . . . . . 7
2625ssriv 3438 . . . . . 6
2726a1i 11 . . . . 5
2820, 27ssexd 4553 . . . 4
2928adantr 467 . . 3
30 eqid 2453 . . . 4
31 eqid 2453 . . . 4
3230, 31isref 20536 . . 3
3329, 32syl 17 . 2
344, 19, 33mpbir2and 934 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   wceq 1446   wcel 1889  cab 2439  wral 2739  wrex 2740  cvv 3047   wss 3406  cpw 3953  csn 3970  cuni 4201   class class class wbr 4405  cref 20529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-ref 20532 This theorem is referenced by:  dispcmp  28698
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