Theorem limenpsi 8135

Description: A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Hypothesis

Ref	Expression
limenpsi.1	|- Lim A

Assertion

Ref	Expression
limenpsi	|- ( A e. V -> A ~~ ( A \ { (/) } ) )

Proof of Theorem limenpsi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difexg 4808	. . 3 |- ( A e. V -> ( A \ { (/) } ) e. _V )
2		limenpsi.1	. . . . . . . 8 |- Lim A
3		limsuc 7049	. . . . . . . 8 |- ( Lim A -> ( x e. A <-> suc x e. A ) )
4	2, 3	ax-mp 5	. . . . . . 7 |- ( x e. A <-> suc x e. A )
5	4	biimpi 206	. . . . . 6 |- ( x e. A -> suc x e. A )
6		nsuceq0 5805	. . . . . 6 |- suc x =/= (/)
7	5, 6	jctir 561	. . . . 5 |- ( x e. A -> ( suc x e. A /\ suc x =/= (/) ) )
8		eldifsn 4317	. . . . 5 |- ( suc x e. ( A \ { (/) } ) <-> ( suc x e. A /\ suc x =/= (/) ) )
9	7, 8	sylibr 224	. . . 4 |- ( x e. A -> suc x e. ( A \ { (/) } ) )
10		limord 5784	. . . . . . 7 |- ( Lim A -> Ord A )
11	2, 10	ax-mp 5	. . . . . 6 |- Ord A
12		ordelon 5747	. . . . . 6 |- ( ( Ord A /\ x e. A ) -> x e. On )
13	11, 12	mpan 706	. . . . 5 |- ( x e. A -> x e. On )
14		ordelon 5747	. . . . . 6 |- ( ( Ord A /\ y e. A ) -> y e. On )
15	11, 14	mpan 706	. . . . 5 |- ( y e. A -> y e. On )
16		suc11 5831	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( suc x = suc y <-> x = y ) )
17	13, 15, 16	syl2an 494	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( suc x = suc y <-> x = y ) )
18	9, 17	dom3 7999	. . 3 |- ( ( A e. V /\ ( A \ { (/) } ) e. _V ) -> A ~<_ ( A \ { (/) } ) )
19	1, 18	mpdan 702	. 2 |- ( A e. V -> A ~<_ ( A \ { (/) } ) )
20		difss 3737	. . 3 |- ( A \ { (/) } ) C_ A
21		ssdomg 8001	. . 3 |- ( A e. V -> ( ( A \ { (/) } ) C_ A -> ( A \ { (/) } ) ~<_ A ) )
22	20, 21	mpi 20	. 2 |- ( A e. V -> ( A \ { (/) } ) ~<_ A )
23		sbth 8080	. 2 |- ( ( A ~<_ ( A \ { (/) } ) /\ ( A \ { (/) } ) ~<_ A ) -> A ~~ ( A \ { (/) } ) )
24	19, 22, 23	syl2anc 693	1 |- ( A e. V -> A ~~ ( A \ { (/) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ~~ cen 7952   ~<_ cdom 7953

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  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-en 7956  df-dom 7957

This theorem is referenced by:  limensuci  8136  omenps  8552  infdifsn  8554  ominf4  9134