Theorem trclfvg 13756

Description: The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.)

Assertion

Ref	Expression
trclfvg	|- ( R C_ ( t+ ` R ) \/ ( t+ ` R ) = (/) )

Proof of Theorem trclfvg

Step	Hyp	Ref	Expression
1		exmid 431	. 2 |- ( R e. _V \/ -. R e. _V )
2		trclfvlb 13749	. . 3 |- ( R e. _V -> R C_ ( t+ ` R ) )
3		fvprc 6185	. . 3 |- ( -. R e. _V -> ( t+ ` R ) = (/) )
4	2, 3	orim12i 538	. 2 |- ( ( R e. _V \/ -. R e. _V ) -> ( R C_ ( t+ ` R ) \/ ( t+ ` R ) = (/) ) )
5	1, 4	ax-mp 5	1 |- ( R C_ ( t+ ` R ) \/ ( t+ ` R ) = (/) )

Syntax hints:   -. wn 3   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ` cfv 5888   t+ctcl 13724

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-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-trcl 13726

This theorem is referenced by: (None)