Theorem brrpssg 6939

Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
brrpssg	|- ( B e. V -> ( A [ C.] B <-> A C. B ) )

Proof of Theorem brrpssg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( B e. V -> B e. _V )
2		relrpss 6938	. . . 4 |- Rel [ C.]
3	2	brrelexi 5158	. . 3 |- ( A [ C.] B -> A e. _V )
4	1, 3	anim12i 590	. 2 |- ( ( B e. V /\ A [ C.] B ) -> ( B e. _V /\ A e. _V ) )
5	1	adantr 481	. . 3 |- ( ( B e. V /\ A C. B ) -> B e. _V )
6		pssss 3702	. . . 4 |- ( A C. B -> A C_ B )
7		ssexg 4804	. . . 4 |- ( ( A C_ B /\ B e. _V ) -> A e. _V )
8	6, 1, 7	syl2anr 495	. . 3 |- ( ( B e. V /\ A C. B ) -> A e. _V )
9	5, 8	jca 554	. 2 |- ( ( B e. V /\ A C. B ) -> ( B e. _V /\ A e. _V ) )
10		psseq1 3694	. . . 4 |- ( x = A -> ( x C. y <-> A C. y ) )
11		psseq2 3695	. . . 4 |- ( y = B -> ( A C. y <-> A C. B ) )
12		df-rpss 6937	. . . 4 |- [ C.] = { <. x , y >. | x C. y }
13	10, 11, 12	brabg 4994	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A [ C.] B <-> A C. B ) )
14	13	ancoms 469	. 2 |- ( ( B e. _V /\ A e. _V ) -> ( A [ C.] B <-> A C. B ) )
15	4, 9, 14	pm5.21nd 941	1 |- ( B e. V -> ( A [ C.] B <-> A C. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   C. wpss 3575   class class class wbr 4653   [ C.] crpss 6936

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-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-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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-rpss 6937

This theorem is referenced by:  brrpss  6940  sorpssi  6943