Theorem pr1or2 7355

Description: An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)

Assertion

Ref	Expression
pr1or2	|- ( ( A e. C /\ B e. D /\ DECID A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )

Proof of Theorem pr1or2

Step	Hyp	Ref	Expression
1		dcne 2411	. . 3 |- (DECID A = B <-> ( A = B \/ A =/= B ) )
2		enpr1g 6940	. . . . . . 7 |- ( A e. C -> { A , A } ~~ 1o )
3	2	ad2antrr 488	. . . . . 6 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> { A , A } ~~ 1o )
4		preq2 3744	. . . . . . . 8 |- ( A = B -> { A , A } = { A , B } )
5	4	breq1d 4092	. . . . . . 7 |- ( A = B -> ( { A , A } ~~ 1o <-> { A , B } ~~ 1o ) )
6	5	adantl 277	. . . . . 6 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> ( { A , A } ~~ 1o <-> { A , B } ~~ 1o ) )
7	3, 6	mpbid 147	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> { A , B } ~~ 1o )
8	7	orcd 738	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
9		pr2ne 7353	. . . . . 6 |- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) )
10	9	biimpar 297	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ A =/= B ) -> { A , B } ~~ 2o )
11	10	olcd 739	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A =/= B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
12	8, 11	jaodan 802	. . 3 |- ( ( ( A e. C /\ B e. D ) /\ ( A = B \/ A =/= B ) ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
13	1, 12	sylan2b 287	. 2 |- ( ( ( A e. C /\ B e. D ) /\ DECID A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
14	13	3impa 1218	1 |- ( ( A e. C /\ B e. D /\ DECID A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   {cpr 3667   class class class wbr 4082   1oc1o 6545   2oc2o 6546   ~~ cen 6875

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-1o 6552  df-2o 6553  df-er 6670  df-en 6878

This theorem is referenced by:  upgr1elem1  15905