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Theorem eltopspOLD 16983
 Description: Construct a topological space from a topology and vice-versa. We say that is a topology on . (This could be proved more efficiently from istpsOLD 16985, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eltopspOLD

Proof of Theorem eltopspOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4213 . . . 4
2 relopab 5001 . . . . . 6
3 df-topspOLD 16964 . . . . . . 7
43releqi 4960 . . . . . 6
52, 4mpbir 201 . . . . 5
65brrelexi 4918 . . . 4
71, 6sylbir 205 . . 3
8 uniexb 4752 . . . 4
97, 8sylibr 204 . . 3
107, 9jca 519 . 2
11 uniexg 4706 . . 3
12 elex 2964 . . 3
1311, 12jca 519 . 2
14 eqeq1 2442 . . . . 5
1514anbi2d 685 . . . 4
16 eleq1 2496 . . . . 5
17 unieq 4024 . . . . . 6
1817eqeq2d 2447 . . . . 5
1916, 18anbi12d 692 . . . 4
2015, 19opelopabg 4473 . . 3
213eleq2i 2500 . . 3
22 eqid 2436 . . . 4
2322biantru 492 . . 3
2420, 21, 233bitr4g 280 . 2
2510, 13, 24pm5.21nii 343 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2956  cop 3817  cuni 4015   class class class wbr 4212  copab 4265   wrel 4883  ctop 16958  ctpsOLD 16960 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-topspOLD 16964
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