Theorem otsndisj 4979

Description: The singletons consisting of ordered triples which have distinct third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.)

Assertion

Ref	Expression
otsndisj	|- ( ( A e. X /\ B e. Y ) -> Disj c e. V { <. A , B , c >. } )

Distinct variable groups:   A, c   B, c   V, c   X, c   Y, c

Proof of Theorem otsndisj

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		otthg 4954	. . . . . . . . . . . 12 |- ( ( A e. X /\ B e. Y /\ c e. V ) -> ( <. A , B , c >. = <. A , B , d >. <-> ( A = A /\ B = B /\ c = d ) ) )
2	1	3expa 1265	. . . . . . . . . . 11 |- ( ( ( A e. X /\ B e. Y ) /\ c e. V ) -> ( <. A , B , c >. = <. A , B , d >. <-> ( A = A /\ B = B /\ c = d ) ) )
3		simp3 1063	. . . . . . . . . . 11 |- ( ( A = A /\ B = B /\ c = d ) -> c = d )
4	2, 3	syl6bi 243	. . . . . . . . . 10 |- ( ( ( A e. X /\ B e. Y ) /\ c e. V ) -> ( <. A , B , c >. = <. A , B , d >. -> c = d ) )
5	4	con3rr3 151	. . . . . . . . 9 |- ( -. c = d -> ( ( ( A e. X /\ B e. Y ) /\ c e. V ) -> -. <. A , B , c >. = <. A , B , d >. ) )
6	5	imp 445	. . . . . . . 8 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ c e. V ) ) -> -. <. A , B , c >. = <. A , B , d >. )
7	6	neqned 2801	. . . . . . 7 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ c e. V ) ) -> <. A , B , c >. =/= <. A , B , d >. )
8		disjsn2 4247	. . . . . . 7 |- ( <. A , B , c >. =/= <. A , B , d >. -> ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) )
9	7, 8	syl 17	. . . . . 6 |- ( ( -. c = d /\ ( ( A e. X /\ B e. Y ) /\ c e. V ) ) -> ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) )
10	9	expcom 451	. . . . 5 |- ( ( ( A e. X /\ B e. Y ) /\ c e. V ) -> ( -. c = d -> ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) ) )
11	10	orrd 393	. . . 4 |- ( ( ( A e. X /\ B e. Y ) /\ c e. V ) -> ( c = d \/ ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) ) )
12	11	adantrr 753	. . 3 |- ( ( ( A e. X /\ B e. Y ) /\ ( c e. V /\ d e. V ) ) -> ( c = d \/ ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) ) )
13	12	ralrimivva 2971	. 2 |- ( ( A e. X /\ B e. Y ) -> A. c e. V A. d e. V ( c = d \/ ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) ) )
14		oteq3 4413	. . . 4 |- ( c = d -> <. A , B , c >. = <. A , B , d >. )
15	14	sneqd 4189	. . 3 |- ( c = d -> { <. A , B , c >. } = { <. A , B , d >. } )
16	15	disjor 4634	. 2 |- (Disj c e. V { <. A , B , c >. } <-> A. c e. V A. d e. V ( c = d \/ ( { <. A , B , c >. } i^i { <. A , B , d >. } ) = (/) ) )
17	13, 16	sylibr 224	1 |- ( ( A e. X /\ B e. Y ) -> Disj c e. V { <. A , B , c >. } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   (/)c0 3915   {csn 4177   <.cotp 4185  Disj wdisj 4620

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-disj 4621

This theorem is referenced by: (None)