Theorem ssprsseq 4357

Description: A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)

Assertion

Ref	Expression
ssprsseq	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> { A , B } = { C , D } ) )

Proof of Theorem ssprsseq

Step	Hyp	Ref	Expression
1		ssprss 4356	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
2	1	3adant3 1081	. . 3 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
3		eqneqall 2805	. . . . . . . 8 |- ( A = B -> ( A =/= B -> { A , B } = { C , D } ) )
4		eqtr3 2643	. . . . . . . 8 |- ( ( A = C /\ B = C ) -> A = B )
5	3, 4	syl11 33	. . . . . . 7 |- ( A =/= B -> ( ( A = C /\ B = C ) -> { A , B } = { C , D } ) )
6	5	3ad2ant3 1084	. . . . . 6 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( A = C /\ B = C ) -> { A , B } = { C , D } ) )
7	6	com12 32	. . . . 5 |- ( ( A = C /\ B = C ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
8		preq12 4270	. . . . . . 7 |- ( ( A = D /\ B = C ) -> { A , B } = { D , C } )
9		prcom 4267	. . . . . . 7 |- { D , C } = { C , D }
10	8, 9	syl6eq 2672	. . . . . 6 |- ( ( A = D /\ B = C ) -> { A , B } = { C , D } )
11	10	a1d 25	. . . . 5 |- ( ( A = D /\ B = C ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
12		preq12 4270	. . . . . 6 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
13	12	a1d 25	. . . . 5 |- ( ( A = C /\ B = D ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
14		eqtr3 2643	. . . . . . . 8 |- ( ( A = D /\ B = D ) -> A = B )
15	3, 14	syl11 33	. . . . . . 7 |- ( A =/= B -> ( ( A = D /\ B = D ) -> { A , B } = { C , D } ) )
16	15	3ad2ant3 1084	. . . . . 6 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( A = D /\ B = D ) -> { A , B } = { C , D } ) )
17	16	com12 32	. . . . 5 |- ( ( A = D /\ B = D ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
18	7, 11, 13, 17	ccase 987	. . . 4 |- ( ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
19	18	com12 32	. . 3 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) -> { A , B } = { C , D } ) )
20	2, 19	sylbid 230	. 2 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } -> { A , B } = { C , D } ) )
21		eqimss 3657	. 2 |- ( { A , B } = { C , D } -> { A , B } C_ { C , D } )
22	20, 21	impbid1 215	1 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> { A , B } = { C , D } ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {cpr 4179

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

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-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  upgredgpr  26037