Theorem bnj563 30813

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj563.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj563.21	|- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )

Assertion

Ref	Expression
bnj563	|- ( ( et /\ rh ) -> suc i e. m )

Proof of Theorem bnj563

Step	Hyp	Ref	Expression
1		bnj563.19	. . 3 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
2		bnj312 30778	. . . . 5 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) <-> ( n = suc m /\ m e. D /\ p e. om /\ m = suc p ) )
3		bnj252 30769	. . . . 5 |- ( ( n = suc m /\ m e. D /\ p e. om /\ m = suc p ) <-> ( n = suc m /\ ( m e. D /\ p e. om /\ m = suc p ) ) )
4	2, 3	bitri 264	. . . 4 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) <-> ( n = suc m /\ ( m e. D /\ p e. om /\ m = suc p ) ) )
5	4	simplbi 476	. . 3 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) -> n = suc m )
6	1, 5	sylbi 207	. 2 |- ( et -> n = suc m )
7		bnj563.21	. . . 4 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
8	7	simp2bi 1077	. . 3 |- ( rh -> suc i e. n )
9	7	simp3bi 1078	. . 3 |- ( rh -> m =/= suc i )
10	8, 9	jca 554	. 2 |- ( rh -> ( suc i e. n /\ m =/= suc i ) )
11		necom 2847	. . . 4 |- ( m =/= suc i <-> suc i =/= m )
12		eleq2 2690	. . . . . 6 |- ( n = suc m -> ( suc i e. n <-> suc i e. suc m ) )
13	12	biimpa 501	. . . . 5 |- ( ( n = suc m /\ suc i e. n ) -> suc i e. suc m )
14		elsuci 5791	. . . . . . 7 |- ( suc i e. suc m -> ( suc i e. m \/ suc i = m ) )
15		orcom 402	. . . . . . . 8 |- ( ( suc i = m \/ suc i e. m ) <-> ( suc i e. m \/ suc i = m ) )
16		neor 2885	. . . . . . . 8 |- ( ( suc i = m \/ suc i e. m ) <-> ( suc i =/= m -> suc i e. m ) )
17	15, 16	bitr3i 266	. . . . . . 7 |- ( ( suc i e. m \/ suc i = m ) <-> ( suc i =/= m -> suc i e. m ) )
18	14, 17	sylib 208	. . . . . 6 |- ( suc i e. suc m -> ( suc i =/= m -> suc i e. m ) )
19	18	imp 445	. . . . 5 |- ( ( suc i e. suc m /\ suc i =/= m ) -> suc i e. m )
20	13, 19	stoic3 1701	. . . 4 |- ( ( n = suc m /\ suc i e. n /\ suc i =/= m ) -> suc i e. m )
21	11, 20	syl3an3b 1364	. . 3 |- ( ( n = suc m /\ suc i e. n /\ m =/= suc i ) -> suc i e. m )
22	21	3expb 1266	. 2 |- ( ( n = suc m /\ ( suc i e. n /\ m =/= suc i ) ) -> suc i e. m )
23	6, 10, 22	syl2an 494	1 |- ( ( et /\ rh ) -> suc i e. m )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   suc csuc 5725   omcom 7065   /\ w-bnj17 30752

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-suc 5729  df-bnj17 30753

This theorem is referenced by:  bnj570  30975