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Metamath Proof Explorer

Theorem limenpsi

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)

Ref Expression
Hypothesis limenpsi.1 
|- Lim A
Assertion limenpsi 
|- ( A e. V -> A ~~ ( A \ { (/) } ) )

Proof

Step Hyp Ref Expression
1 limenpsi.1 
 |-  Lim A
2 difexg 
 |-  ( A e. V -> ( A \ { (/) } ) e. _V )
3 limsuc 
 |-  ( Lim A -> ( x e. A <-> suc x e. A ) )
4 1 3 ax-mp 
 |-  ( x e. A <-> suc x e. A )
5 4 biimpi 
 |-  ( x e. A -> suc x e. A )
6 nsuceq0 
 |-  suc x =/= (/)
7 eldifsn 
 |-  ( suc x e. ( A \ { (/) } ) <-> ( suc x e. A /\ suc x =/= (/) ) )
8 5 6 7 sylanblrc 
 |-  ( x e. A -> suc x e. ( A \ { (/) } ) )
9 limord 
 |-  ( Lim A -> Ord A )
10 1 9 ax-mp 
 |-  Ord A
11 ordelon 
 |-  ( ( Ord A /\ x e. A ) -> x e. On )
12 10 11 mpan 
 |-  ( x e. A -> x e. On )
13 ordelon 
 |-  ( ( Ord A /\ y e. A ) -> y e. On )
14 10 13 mpan 
 |-  ( y e. A -> y e. On )
15 suc11 
 |-  ( ( x e. On /\ y e. On ) -> ( suc x = suc y <-> x = y ) )
16 12 14 15 syl2an 
 |-  ( ( x e. A /\ y e. A ) -> ( suc x = suc y <-> x = y ) )
17 8 16 dom3 
 |-  ( ( A e. V /\ ( A \ { (/) } ) e. _V ) -> A ~<_ ( A \ { (/) } ) )
18 2 17 mpdan 
 |-  ( A e. V -> A ~<_ ( A \ { (/) } ) )
19 difss 
 |-  ( A \ { (/) } ) C_ A
20 ssdomg 
 |-  ( A e. V -> ( ( A \ { (/) } ) C_ A -> ( A \ { (/) } ) ~<_ A ) )
21 19 20 mpi 
 |-  ( A e. V -> ( A \ { (/) } ) ~<_ A )
22 sbth 
 |-  ( ( A ~<_ ( A \ { (/) } ) /\ ( A \ { (/) } ) ~<_ A ) -> A ~~ ( A \ { (/) } ) )
23 18 21 22 syl2anc 
 |-  ( A e. V -> A ~~ ( A \ { (/) } ) )