Theorem pr1or2 7399

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 2413	. . 3 |- (DECID A = B <-> ( A = B \/ A =/= B ) )
2		enpr1g 6972	. . . . . . 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 3749	. . . . . . . 8 |- ( A = B -> { A , A } = { A , B } )
5	4	breq1d 4098	. . . . . . 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 740	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
9		pr2ne 7397	. . . . . 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 741	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A =/= B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
12	8, 11	jaodan 804	. . 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 1220	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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   {cpr 3670   class class class wbr 4088   1oc1o 6575   2oc2o 6576   ~~ cen 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910

This theorem is referenced by:  upgr1elem1  15977