Theorem 3elpr2eq 4435

Description: If there are three elements in a proper unordered pair, and two of them are different from the third one, the two must be equal. (Contributed by AV, 19-Dec-2021.)

Assertion

Ref	Expression
3elpr2eq	|- ( ( ( X e. { A , B } /\ Y e. { A , B } /\ Z e. { A , B } ) /\ ( Y =/= X /\ Z =/= X ) ) -> Y = Z )

Proof of Theorem 3elpr2eq

Step	Hyp	Ref	Expression
1		elpri 4197	. . 3 |- ( X e. { A , B } -> ( X = A \/ X = B ) )
2		elpri 4197	. . 3 |- ( Y e. { A , B } -> ( Y = A \/ Y = B ) )
3		elpri 4197	. . 3 |- ( Z e. { A , B } -> ( Z = A \/ Z = B ) )
4		eqtr3 2643	. . . . . . . . . . 11 |- ( ( Z = A /\ X = A ) -> Z = X )
5		eqneqall 2805	. . . . . . . . . . 11 |- ( Z = X -> ( Z =/= X -> Y = Z ) )
6	4, 5	syl 17	. . . . . . . . . 10 |- ( ( Z = A /\ X = A ) -> ( Z =/= X -> Y = Z ) )
7	6	adantld 483	. . . . . . . . 9 |- ( ( Z = A /\ X = A ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) )
8	7	ex 450	. . . . . . . 8 |- ( Z = A -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) )
9	8	a1d 25	. . . . . . 7 |- ( Z = A -> ( ( Y = A \/ Y = B ) -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
10		eqtr3 2643	. . . . . . . . . . . . 13 |- ( ( Y = A /\ X = A ) -> Y = X )
11		eqneqall 2805	. . . . . . . . . . . . 13 |- ( Y = X -> ( Y =/= X -> ( Z =/= X -> Y = Z ) ) )
12	10, 11	syl 17	. . . . . . . . . . . 12 |- ( ( Y = A /\ X = A ) -> ( Y =/= X -> ( Z =/= X -> Y = Z ) ) )
13	12	impd 447	. . . . . . . . . . 11 |- ( ( Y = A /\ X = A ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) )
14	13	ex 450	. . . . . . . . . 10 |- ( Y = A -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) )
15	14	a1d 25	. . . . . . . . 9 |- ( Y = A -> ( Z = B -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
16		eqtr3 2643	. . . . . . . . . . 11 |- ( ( Y = B /\ Z = B ) -> Y = Z )
17	16	2a1d 26	. . . . . . . . . 10 |- ( ( Y = B /\ Z = B ) -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) )
18	17	ex 450	. . . . . . . . 9 |- ( Y = B -> ( Z = B -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
19	15, 18	jaoi 394	. . . . . . . 8 |- ( ( Y = A \/ Y = B ) -> ( Z = B -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
20	19	com12 32	. . . . . . 7 |- ( Z = B -> ( ( Y = A \/ Y = B ) -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
21	9, 20	jaoi 394	. . . . . 6 |- ( ( Z = A \/ Z = B ) -> ( ( Y = A \/ Y = B ) -> ( X = A -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
22	21	com13 88	. . . . 5 |- ( X = A -> ( ( Y = A \/ Y = B ) -> ( ( Z = A \/ Z = B ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
23		eqtr3 2643	. . . . . . . . . . 11 |- ( ( Y = A /\ Z = A ) -> Y = Z )
24	23	2a1d 26	. . . . . . . . . 10 |- ( ( Y = A /\ Z = A ) -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) )
25	24	ex 450	. . . . . . . . 9 |- ( Y = A -> ( Z = A -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
26		eqtr3 2643	. . . . . . . . . . . . 13 |- ( ( Y = B /\ X = B ) -> Y = X )
27	26, 11	syl 17	. . . . . . . . . . . 12 |- ( ( Y = B /\ X = B ) -> ( Y =/= X -> ( Z =/= X -> Y = Z ) ) )
28	27	impd 447	. . . . . . . . . . 11 |- ( ( Y = B /\ X = B ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) )
29	28	ex 450	. . . . . . . . . 10 |- ( Y = B -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) )
30	29	a1d 25	. . . . . . . . 9 |- ( Y = B -> ( Z = A -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
31	25, 30	jaoi 394	. . . . . . . 8 |- ( ( Y = A \/ Y = B ) -> ( Z = A -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
32	31	com12 32	. . . . . . 7 |- ( Z = A -> ( ( Y = A \/ Y = B ) -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
33		eqtr3 2643	. . . . . . . . . . 11 |- ( ( Z = B /\ X = B ) -> Z = X )
34	33, 5	syl 17	. . . . . . . . . 10 |- ( ( Z = B /\ X = B ) -> ( Z =/= X -> Y = Z ) )
35	34	adantld 483	. . . . . . . . 9 |- ( ( Z = B /\ X = B ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) )
36	35	ex 450	. . . . . . . 8 |- ( Z = B -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) )
37	36	a1d 25	. . . . . . 7 |- ( Z = B -> ( ( Y = A \/ Y = B ) -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
38	32, 37	jaoi 394	. . . . . 6 |- ( ( Z = A \/ Z = B ) -> ( ( Y = A \/ Y = B ) -> ( X = B -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
39	38	com13 88	. . . . 5 |- ( X = B -> ( ( Y = A \/ Y = B ) -> ( ( Z = A \/ Z = B ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
40	22, 39	jaoi 394	. . . 4 |- ( ( X = A \/ X = B ) -> ( ( Y = A \/ Y = B ) -> ( ( Z = A \/ Z = B ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) ) ) )
41	40	3imp 1256	. . 3 |- ( ( ( X = A \/ X = B ) /\ ( Y = A \/ Y = B ) /\ ( Z = A \/ Z = B ) ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) )
42	1, 2, 3, 41	syl3an 1368	. 2 |- ( ( X e. { A , B } /\ Y e. { A , B } /\ Z e. { A , B } ) -> ( ( Y =/= X /\ Z =/= X ) -> Y = Z ) )
43	42	imp 445	1 |- ( ( ( X e. { A , B } /\ Y e. { A , B } /\ Z e. { A , B } ) /\ ( Y =/= X /\ Z =/= X ) ) -> Y = Z )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  numedglnl  26039