Metamath Proof Explorer


Theorem inaex

Description: Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023)

Ref Expression
Assertion inaex
|- ( A e. On -> E. x e. Inacc A e. x )

Proof

Step Hyp Ref Expression
1 inawina
 |-  ( x e. Inacc -> x e. InaccW )
2 winaon
 |-  ( x e. InaccW -> x e. On )
3 1 2 syl
 |-  ( x e. Inacc -> x e. On )
4 3 ssriv
 |-  Inacc C_ On
5 onmindif
 |-  ( ( Inacc C_ On /\ A e. On ) -> A e. |^| ( Inacc \ suc A ) )
6 4 5 mpan
 |-  ( A e. On -> A e. |^| ( Inacc \ suc A ) )
7 6 adantr
 |-  ( ( A e. On /\ x = |^| ( Inacc \ suc A ) ) -> A e. |^| ( Inacc \ suc A ) )
8 simpr
 |-  ( ( A e. On /\ x = |^| ( Inacc \ suc A ) ) -> x = |^| ( Inacc \ suc A ) )
9 7 8 eleqtrrd
 |-  ( ( A e. On /\ x = |^| ( Inacc \ suc A ) ) -> A e. x )
10 difss
 |-  ( Inacc \ suc A ) C_ Inacc
11 10 4 sstri
 |-  ( Inacc \ suc A ) C_ On
12 inaprc
 |-  Inacc e/ _V
13 12 neli
 |-  -. Inacc e. _V
14 ssdif0
 |-  ( Inacc C_ suc A <-> ( Inacc \ suc A ) = (/) )
15 sucexg
 |-  ( A e. On -> suc A e. _V )
16 ssexg
 |-  ( ( Inacc C_ suc A /\ suc A e. _V ) -> Inacc e. _V )
17 16 expcom
 |-  ( suc A e. _V -> ( Inacc C_ suc A -> Inacc e. _V ) )
18 15 17 syl
 |-  ( A e. On -> ( Inacc C_ suc A -> Inacc e. _V ) )
19 14 18 syl5bir
 |-  ( A e. On -> ( ( Inacc \ suc A ) = (/) -> Inacc e. _V ) )
20 13 19 mtoi
 |-  ( A e. On -> -. ( Inacc \ suc A ) = (/) )
21 20 neqned
 |-  ( A e. On -> ( Inacc \ suc A ) =/= (/) )
22 onint
 |-  ( ( ( Inacc \ suc A ) C_ On /\ ( Inacc \ suc A ) =/= (/) ) -> |^| ( Inacc \ suc A ) e. ( Inacc \ suc A ) )
23 11 21 22 sylancr
 |-  ( A e. On -> |^| ( Inacc \ suc A ) e. ( Inacc \ suc A ) )
24 23 eldifad
 |-  ( A e. On -> |^| ( Inacc \ suc A ) e. Inacc )
25 9 24 rspcime
 |-  ( A e. On -> E. x e. Inacc A e. x )