Theorem onintrab2 7002

Description: An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)

Assertion

Ref	Expression
onintrab2	|- ( E. x e. On ph <-> |^| { x e. On | ph } e. On )

Proof of Theorem onintrab2

Step	Hyp	Ref	Expression
1		intexrab 4823	. 2 |- ( E. x e. On ph <-> |^| { x e. On | ph } e. _V )
2		onintrab 7001	. 2 |- ( |^| { x e. On | ph } e. _V <-> |^| { x e. On | ph } e. On )
3	1, 2	bitri 264	1 |- ( E. x e. On ph <-> |^| { x e. On | ph } e. On )

Syntax hints:   <-> wb 196   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   |^|cint 4475   Oncon0 5723

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-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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  oeeulem  7681  cardmin2  8824  cardaleph  8912  cardmin  9386  nosepon  31818