Theorem oe0m1 7074

Description: Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.)

Assertion

Ref	Expression
oe0m1	|- ( A e. On -> ( (/) e. A <-> ( (/) ^o A ) = (/) ) )

Proof of Theorem oe0m1

Step	Hyp	Ref	Expression
1		eloni 4840	. . 3 |- ( A e. On -> Ord A )
2		ordgt0ge1 7050	. . 3 |- ( Ord A -> ( (/) e. A <-> 1o C_ A ) )
3	1, 2	syl 16	. 2 |- ( A e. On -> ( (/) e. A <-> 1o C_ A ) )
4		oe0m 7071	. . . 4 |- ( A e. On -> ( (/) ^o A ) = ( 1o \ A ) )
5	4	eqeq1d 2456	. . 3 |- ( A e. On -> ( ( (/) ^o A ) = (/) <-> ( 1o \ A ) = (/) ) )
6		ssdif0 3848	. . 3 |- ( 1o C_ A <-> ( 1o \ A ) = (/) )
7	5, 6	syl6rbbr 264	. 2 |- ( A e. On -> ( 1o C_ A <-> ( (/) ^o A ) = (/) ) )
8	3, 7	bitrd 253	1 |- ( A e. On -> ( (/) e. A <-> ( (/) ^o A ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1370   e. wcel 1758   \ cdif 3436   C_ wss 3439   (/)c0 3748   Ord word 4829   Oncon0 4830  (class class class)co 6203   1oc1o 7026   ^o coe 7032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-recs 6945  df-rdg 6979  df-1o 7033  df-oexp 7039

This theorem is referenced by:  oev2  7076  oesuclem  7078  oecl  7090  oewordri  7144  oelim2  7147  oeoa  7149  oeoe  7151  cantnf  8015  cantnfOLD  8037