Theorem relintabex 37887

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 4821	. . . 4 |- ( -. |^| { x | ph } e. _V <-> |^| { x | ph } = _V )
2		0nelxp 5143	. . . . . . 7 |- -. (/) e. ( _V X. _V )
3		0ex 4790	. . . . . . . 8 |- (/) e. _V
4		eleq1 2689	. . . . . . . . 9 |- ( x = (/) -> ( x e. ( _V X. _V ) <-> (/) e. ( _V X. _V ) ) )
5	4	notbid 308	. . . . . . . 8 |- ( x = (/) -> ( -. x e. ( _V X. _V ) <-> -. (/) e. ( _V X. _V ) ) )
6	3, 5	spcev 3300	. . . . . . 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 3663	. . . . . . . 8 |- ( -. _V C_ ( _V X. _V ) <-> E. x ( x e. _V /\ -. x e. ( _V X. _V ) ) )
9		df-rex 2918	. . . . . . . 8 |- ( E. x e. _V -. x e. ( _V X. _V ) <-> E. x ( x e. _V /\ -. x e. ( _V X. _V ) ) )
10		rexv 3220	. . . . . . . 8 |- ( E. x e. _V -. x e. ( _V X. _V ) <-> E. x -. x e. ( _V X. _V ) )
11	8, 9, 10	3bitr2i 288	. . . . . . 7 |- ( -. _V C_ ( _V X. _V ) <-> E. x -. x e. ( _V X. _V ) )
12		df-rel 5121	. . . . . . 7 |- ( Rel _V <-> _V C_ ( _V X. _V ) )
13	11, 12	xchnxbir 323	. . . . . 6 |- ( -. Rel _V <-> E. x -. x e. ( _V X. _V ) )
14	7, 13	mpbir 221	. . . . 5 |- -. Rel _V
15		releq 5201	. . . . 5 |- ( |^| { x | ph } = _V -> ( Rel |^| { x | ph } <-> Rel _V ) )
16	14, 15	mtbiri 317	. . . 4 |- ( |^| { x | ph } = _V -> -. Rel |^| { x | ph } )
17	1, 16	sylbi 207	. . 3 |- ( -. |^| { x | ph } e. _V -> -. Rel |^| { x | ph } )
18	17	con4i 113	. 2 |- ( Rel |^| { x | ph } -> |^| { x | ph } e. _V )
19		intexab 4822	. 2 |- ( E. x ph <-> |^| { x | ph } e. _V )
20	18, 19	sylibr 224	1 |- ( Rel |^| { x | ph } -> E. x ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|cint 4475   X. cxp 5112   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  relintab  37889