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Theorem oe0m0 7162
Description: Ordinal exponentiation with zero mantissa and zero exponent. Proposition 8.31 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.)
Assertion
Ref Expression
oe0m0  |-  ( (/)  ^o  (/) )  =  1o

Proof of Theorem oe0m0
StepHypRef Expression
1 0elon 4920 . 2  |-  (/)  e.  On
2 oe0m 7160 . . 3  |-  ( (/)  e.  On  ->  ( (/)  ^o  (/) )  =  ( 1o  \  (/) ) )
3 dif0 3886 . . 3  |-  ( 1o 
\  (/) )  =  1o
42, 3syl6eq 2511 . 2  |-  ( (/)  e.  On  ->  ( (/)  ^o  (/) )  =  1o )
51, 4ax-mp 5 1  |-  ( (/)  ^o  (/) )  =  1o
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823    \ cdif 3458   (/)c0 3783   Oncon0 4867  (class class class)co 6270   1oc1o 7115    ^o coe 7121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-1o 7122  df-oexp 7128
This theorem is referenced by:  oe0  7164  oev2  7165  oesuclem  7167  oecl  7179  oeoa  7238  oeoe  7240
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