Theorem trint0m 3871

Description: Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)

Assertion

Ref	Expression
trint0m	|- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )

Distinct variable group:   x, A

Proof of Theorem trint0m

Step	Hyp	Ref	Expression
1		intss1 3630	. . . 4 |- ( x e. A -> |^| A C_ x )
2		trss 3863	. . . . 5 |- ( Tr A -> ( x e. A -> x C_ A ) )
3	2	com12 27	. . . 4 |- ( x e. A -> ( Tr A -> x C_ A ) )
4		sstr2 2952	. . . 4 |- ( |^| A C_ x -> ( x C_ A -> |^| A C_ A ) )
5	1, 3, 4	sylsyld 52	. . 3 |- ( x e. A -> ( Tr A -> |^| A C_ A ) )
6	5	exlimiv 1489	. 2 |- ( E. x x e. A -> ( Tr A -> |^| A C_ A ) )
7	6	impcom 116	1 |- ( ( Tr A /\ E. x x e. A ) -> |^| A C_ A )

Syntax hints:   -> wi 4   /\ wa 97   E.wex 1381   e. wcel 1393   C_ wss 2917   |^|cint 3615   Tr wtr 3854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616  df-tr 3855

This theorem is referenced by: (None)