Theorem ssprsseq 3872

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 3871	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
2	1	3adant3 1048	. . 3 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
3		eqneqall 2430	. . . . . . . 8 |- ( A = B -> ( A =/= B -> { A , B } = { C , D } ) )
4		eqtr3 2258	. . . . . . . 8 |- ( ( A = C /\ B = C ) -> A = B )
5	3, 4	syl11 31	. . . . . . 7 |- ( A =/= B -> ( ( A = C /\ B = C ) -> { A , B } = { C , D } ) )
6	5	3ad2ant3 1051	. . . . . 6 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( A = C /\ B = C ) -> { A , B } = { C , D } ) )
7	6	com12 30	. . . . 5 |- ( ( A = C /\ B = C ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
8		preq12 3786	. . . . . . 7 |- ( ( A = D /\ B = C ) -> { A , B } = { D , C } )
9		prcom 3783	. . . . . . 7 |- { D , C } = { C , D }
10	8, 9	eqtrdi 2287	. . . . . 6 |- ( ( A = D /\ B = C ) -> { A , B } = { C , D } )
11	10	a1d 22	. . . . 5 |- ( ( A = D /\ B = C ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
12		preq12 3786	. . . . . 6 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
13	12	a1d 22	. . . . 5 |- ( ( A = C /\ B = D ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
14		eqtr3 2258	. . . . . . . 8 |- ( ( A = D /\ B = D ) -> A = B )
15	3, 14	syl11 31	. . . . . . 7 |- ( A =/= B -> ( ( A = D /\ B = D ) -> { A , B } = { C , D } ) )
16	15	3ad2ant3 1051	. . . . . 6 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( ( A = D /\ B = D ) -> { A , B } = { C , D } ) )
17	16	com12 30	. . . . 5 |- ( ( A = D /\ B = D ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
18	7, 11, 13, 17	ccase 977	. . . 4 |- ( ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) -> ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } = { C , D } ) )
19	18	com12 30	. . 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 150	. 2 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } -> { A , B } = { C , D } ) )
21		eqimss 3302	. 2 |- ( { A , B } = { C , D } -> { A , B } C_ { C , D } )
22	20, 21	impbid1 142	1 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> { A , B } = { C , D } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   C_ wss 3220   {cpr 3706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712

This theorem is referenced by:  upgredgpr  16304