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Theorem tz7.5 4589
Description: A subclass (possibly proper) of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.)
Assertion
Ref Expression
tz7.5  |-  ( ( Ord  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
Distinct variable group:    x, B
Allowed substitution hint:    A( x)

Proof of Theorem tz7.5
StepHypRef Expression
1 ordwe 4581 . 2  |-  ( Ord 
A  ->  _E  We  A )
2 wefrc 4563 . 2  |-  ( (  _E  We  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
31, 2syl3an1 1217 1  |-  ( ( Ord  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  ( B  i^i  x )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    =/= wne 2593   E.wrex 2693    i^i cin 3306    C_ wss 3307   (/)c0 3615    _E cep 4479    We wwe 4527   Ord word 4567
This theorem is referenced by:  tz7.7  4594  onint  4761  tfi  4819  peano5  4854  fin23lem26  8189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-br 4200  df-opab 4254  df-eprel 4481  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571
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