Theorem trint0 4562

Description: Any nonempty transitive class includes its intersection. Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)

Assertion

Ref	Expression
trint0

Proof of Theorem trint0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3794	. . 3
2		intss1 4301	. . . . 5
3		trss 4554	. . . . . 6
4	3	com12 31	. . . . 5
5		sstr2 3510	. . . . 5
6	2, 4, 5	sylsyld 56	. . . 4
7	6	exlimiv 1722	. . 3
8	1, 7	sylbi 195	. 2
9	8	impcom 430	1

Syntax hints:  ->wi 4  /\wa 369  E.wex 1612  e.wcel 1818  =/=wne 2652  C_wss 3475  c0 3784  |^|cint 4286  Trwtr 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-uni 4250  df-int 4287  df-tr 4546