Theorem relintabex 36232

Description: If the intersection of a class is a relation, then the class is non-empty. (Contributed by RP, 12-Aug-2020.)

Assertion

Ref	Expression
relintabex	|- ( Rel |^| { x | ph } -> E. x ph )

Proof of Theorem relintabex

Step	Hyp	Ref	Expression
1		intnex 4574	. . . 4 |- ( -. |^| { x | ph } e. _V <-> |^| { x | ph } = _V )
2		0nelxp 4881	. . . . . . 7 |- -. (/) e. ( _V X. _V )
3		0ex 4549	. . . . . . . 8 |- (/) e. _V
4		eleq1 2528	. . . . . . . . 9 |- ( x = (/) -> ( x e. ( _V X. _V ) <-> (/) e. ( _V X. _V ) ) )
5	4	notbid 300	. . . . . . . 8 |- ( x = (/) -> ( -. x e. ( _V X. _V ) <-> -. (/) e. ( _V X. _V ) ) )
6	3, 5	spcev 3153	. . . . . . 7 |- ( -. (/) e. ( _V X. _V ) -> E. x -. x e. ( _V X. _V ) )
7	2, 6	ax-mp 5	. . . . . 6 |- E. x -. x e. ( _V X. _V )
8		nss 3502	. . . . . . . 8 |- ( -. _V C_ ( _V X. _V ) <-> E. x ( x e. _V /\ -. x e. ( _V X. _V ) ) )
9		df-rex 2755	. . . . . . . 8 |- ( E. x e. _V -. x e. ( _V X. _V ) <-> E. x ( x e. _V /\ -. x e. ( _V X. _V ) ) )
10		rexv 3074	. . . . . . . 8 |- ( E. x e. _V -. x e. ( _V X. _V ) <-> E. x -. x e. ( _V X. _V ) )
11	8, 9, 10	3bitr2i 281	. . . . . . 7 |- ( -. _V C_ ( _V X. _V ) <-> E. x -. x e. ( _V X. _V ) )
12		df-rel 4860	. . . . . . 7 |- ( Rel _V <-> _V C_ ( _V X. _V ) )
13	11, 12	xchnxbir 315	. . . . . 6 |- ( -. Rel _V <-> E. x -. x e. ( _V X. _V ) )
14	7, 13	mpbir 214	. . . . 5 |- -. Rel _V
15		releq 4936	. . . . 5 |- ( |^| { x | ph } = _V -> ( Rel |^| { x | ph } <-> Rel _V ) )
16	14, 15	mtbiri 309	. . . 4 |- ( |^| { x | ph } = _V -> -. Rel |^| { x | ph } )
17	1, 16	sylbi 200	. . 3 |- ( -. |^| { x | ph } e. _V -> -. Rel |^| { x | ph } )
18	17	con4i 135	. 2 |- ( Rel |^| { x | ph } -> |^| { x | ph } e. _V )
19		intexab 4575	. 2 |- ( E. x ph <-> |^| { x | ph } e. _V )
20	18, 19	sylibr 217	1 |- ( Rel |^| { x | ph } -> E. x ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   = wceq 1455   E.wex 1674   e. wcel 1898   {cab 2448   E.wrex 2750   _Vcvv 3057   C_ wss 3416   (/)c0 3743   |^|cint 4248   X. cxp 4851   Rel wrel 4858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-int 4249  df-opab 4476  df-xp 4859  df-rel 4860

This theorem is referenced by:  relintab  36234