Theorem bnj158 30797

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

Hypothesis

Ref	Expression
bnj158.1	|- D = ( om \ { (/) } )

Assertion

Ref	Expression
bnj158	|- ( m e. D -> E. p e. om m = suc p )

Distinct variable group:   m, p

Allowed substitution hints:   D( m, p)

Proof of Theorem bnj158

Step	Hyp	Ref	Expression
1		bnj158.1	. . . 4 |- D = ( om \ { (/) } )
2	1	eleq2i 2693	. . 3 |- ( m e. D <-> m e. ( om \ { (/) } ) )
3		eldifsn 4317	. . 3 |- ( m e. ( om \ { (/) } ) <-> ( m e. om /\ m =/= (/) ) )
4	2, 3	bitri 264	. 2 |- ( m e. D <-> ( m e. om /\ m =/= (/) ) )
5		nnsuc 7082	. 2 |- ( ( m e. om /\ m =/= (/) ) -> E. p e. om m = suc p )
6	4, 5	sylbi 207	1 |- ( m e. D -> E. p e. om m = suc p )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   suc csuc 5725   omcom 7065

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-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-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-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  bnj168  30798  bnj600  30989  bnj986  31024