Theorem distps 7604

Description: The discrete topology on a set A expressed as a topological space. (Contributed by FL, 20-Aug-2006.)

Hypothesis

Ref	Expression
indistop.1	|- A e. V

Assertion

Ref	Expression
distps	|- <.A, P~A>. e. TopSp

Proof of Theorem distps

Step	Hyp	Ref	Expression
1		unipw 2751	. . 3 |- U.P~A = A
2	1	opeq1i 2486	. 2 |- <.U.P~A, P~A>. = <.A, P~A>.
3		indistop.1	. . . 4 |- A e. V
4	3	distop 7599	. . 3 |- P~A e. Top
5		eltopsp 7554	. . 3 |- (<.U.P~A, P~A>. e. TopSp <-> P~A e. Top)
6	4, 5	mpbir 190	. 2 |- <.U.P~A, P~A>. e. TopSp
7	2, 6	eqeltrr 1542	1 |- <.A, P~A>. e. TopSp

Syntax hints:   e. wcel 956  Vcvv 1807  P~cpw 2397  <.cop 2407  U.cuni 2498  Topctop 7538  TopSpctps 7539

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-top 7542  df-topsp 7543

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