Theorem pr1or2 7490

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 2423	. . 3 |- (DECID A = B <-> ( A = B \/ A =/= B ) )
2		enpr1g 7037	. . . . . . 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 3768	. . . . . . . 8 |- ( A = B -> { A , A } = { A , B } )
5	4	breq1d 4118	. . . . . . 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 741	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
9		pr2ne 7488	. . . . . 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 742	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A =/= B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
12	8, 11	jaodan 805	. . 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 1221	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   {cpr 3689   class class class wbr 4108   1oc1o 6639   2oc2o 6640   ~~ cen 6972

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975

This theorem is referenced by:  upgr1elem1  16107