Theorem pr2ne 8828

Description: If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)

Assertion

Ref	Expression
pr2ne	|- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) )

Proof of Theorem pr2ne

Step	Hyp	Ref	Expression
1		preq2 4269	. . . . 5 |- ( B = A -> { A , B } = { A , A } )
2	1	eqcoms 2630	. . . 4 |- ( A = B -> { A , B } = { A , A } )
3		enpr1g 8022	. . . . . . . 8 |- ( A e. C -> { A , A } ~~ 1o )
4		entr 8008	. . . . . . . . . . . 12 |- ( ( { A , B } ~~ { A , A } /\ { A , A } ~~ 1o ) -> { A , B } ~~ 1o )
5		1sdom2 8159	. . . . . . . . . . . . . . 15 |- 1o ~< 2o
6		sdomnen 7984	. . . . . . . . . . . . . . 15 |- ( 1o ~< 2o -> -. 1o ~~ 2o )
7	5, 6	ax-mp 5	. . . . . . . . . . . . . 14 |- -. 1o ~~ 2o
8		ensym 8005	. . . . . . . . . . . . . . 15 |- ( { A , B } ~~ 1o -> 1o ~~ { A , B } )
9		entr 8008	. . . . . . . . . . . . . . . 16 |- ( ( 1o ~~ { A , B } /\ { A , B } ~~ 2o ) -> 1o ~~ 2o )
10	9	ex 450	. . . . . . . . . . . . . . 15 |- ( 1o ~~ { A , B } -> ( { A , B } ~~ 2o -> 1o ~~ 2o ) )
11	8, 10	syl 17	. . . . . . . . . . . . . 14 |- ( { A , B } ~~ 1o -> ( { A , B } ~~ 2o -> 1o ~~ 2o ) )
12	7, 11	mtoi 190	. . . . . . . . . . . . 13 |- ( { A , B } ~~ 1o -> -. { A , B } ~~ 2o )
13	12	a1d 25	. . . . . . . . . . . 12 |- ( { A , B } ~~ 1o -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) )
14	4, 13	syl 17	. . . . . . . . . . 11 |- ( ( { A , B } ~~ { A , A } /\ { A , A } ~~ 1o ) -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) )
15	14	ex 450	. . . . . . . . . 10 |- ( { A , B } ~~ { A , A } -> ( { A , A } ~~ 1o -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) ) )
16		prex 4909	. . . . . . . . . . 11 |- { A , B } e. _V
17		eqeng 7989	. . . . . . . . . . 11 |- ( { A , B } e. _V -> ( { A , B } = { A , A } -> { A , B } ~~ { A , A } ) )
18	16, 17	ax-mp 5	. . . . . . . . . 10 |- ( { A , B } = { A , A } -> { A , B } ~~ { A , A } )
19	15, 18	syl11 33	. . . . . . . . 9 |- ( { A , A } ~~ 1o -> ( { A , B } = { A , A } -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) ) )
20	19	a1dd 50	. . . . . . . 8 |- ( { A , A } ~~ 1o -> ( { A , B } = { A , A } -> ( B e. D -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) ) ) )
21	3, 20	syl 17	. . . . . . 7 |- ( A e. C -> ( { A , B } = { A , A } -> ( B e. D -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) ) ) )
22	21	com23 86	. . . . . 6 |- ( A e. C -> ( B e. D -> ( { A , B } = { A , A } -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) ) ) )
23	22	imp 445	. . . . 5 |- ( ( A e. C /\ B e. D ) -> ( { A , B } = { A , A } -> ( ( A e. C /\ B e. D ) -> -. { A , B } ~~ 2o ) ) )
24	23	pm2.43a 54	. . . 4 |- ( ( A e. C /\ B e. D ) -> ( { A , B } = { A , A } -> -. { A , B } ~~ 2o ) )
25	2, 24	syl5 34	. . 3 |- ( ( A e. C /\ B e. D ) -> ( A = B -> -. { A , B } ~~ 2o ) )
26	25	necon2ad 2809	. 2 |- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o -> A =/= B ) )
27		pr2nelem 8827	. . 3 |- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o )
28	27	3expia 1267	. 2 |- ( ( A e. C /\ B e. D ) -> ( A =/= B -> { A , B } ~~ 2o ) )
29	26, 28	impbid 202	1 |- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {cpr 4179   class class class wbr 4653   1oc1o 7553   2oc2o 7554   ~~ cen 7952   ~< csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  prdom2  8829  isprm2lem  15394  pmtrrn2  17880  mdetunilem7  20424