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Theorem ordn2lp 4384
Description: An ordinal class cannot an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
ordn2lp  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)

Proof of Theorem ordn2lp
StepHypRef Expression
1 ordirr 4382 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 ordtr 4378 . . 3  |-  ( Ord 
A  ->  Tr  A
)
3 trel 4094 . . 3  |-  ( Tr  A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
42, 3syl 17 . 2  |-  ( Ord 
A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
51, 4mtod 170 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    e. wcel 1621   Tr wtr 4087   Ord word 4363
This theorem is referenced by:  ordtri1  4397  ordnbtwn  4455  suc11  4468  smoord  6350  unblem1  7077  cantnfp1lem3  7350  cardprclem  7580
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-fr 4324  df-we 4326  df-ord 4367
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