Theorem pr1or2 7313

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 2388	. . 3 |- (DECID A = B <-> ( A = B \/ A =/= B ) )
2		enpr1g 6900	. . . . . . 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 3713	. . . . . . . 8 |- ( A = B -> { A , A } = { A , B } )
5	4	breq1d 4058	. . . . . . 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 735	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
9		pr2ne 7312	. . . . . 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 736	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ A =/= B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
12	8, 11	jaodan 799	. . 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 1197	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   {cpr 3636   class class class wbr 4048   1oc1o 6505   2oc2o 6506   ~~ cen 6835

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-1o 6512  df-2o 6513  df-er 6630  df-en 6838

This theorem is referenced by:  upgr1elem1  15763