Theorem prnzg 14454

Description: A pair containing a set is not empty.

Assertion

Ref	Expression
prnzg	|- (A e. C -> {A, B} =/= (/))

Proof of Theorem prnzg

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (A e. C -> A e. _V)
2		preq1 3098	. . . 4 |- (A = if(A e. _V, A, (/)) -> {A, B} = {if(A e. _V, A, (/)), B})
3	2	neeq1d 2028	. . 3 |- (A = if(A e. _V, A, (/)) -> ({A, B} =/= (/) <-> {if(A e. _V, A, (/)), B} =/= (/)))
4		0ex 3446	. . . . 5 |- (/) e. _V
5	4	elimel 3025	. . . 4 |- if(A e. _V, A, (/)) e. _V
6	5	prnz 3120	. . 3 |- {if(A e. _V, A, (/)), B} =/= (/)
7	3, 6	dedth 3011	. 2 |- (A e. _V -> {A, B} =/= (/))
8	1, 7	syl 12	1 |- (A e. C -> {A, B} =/= (/))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875  ifcif 2982  {cpr 3045

This theorem is referenced by:  inttop4 14865  pmapmeet 17255

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-nul 2876  df-if 2983  df-sn 3049  df-pr 3050

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