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Theorem om0x 8002
 Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8000, this version works whether or not 𝐴 is an ordinal. However, since it is an artifact of our particular function value definition outside the domain, we will not use it in order to be conventional and present it only as a curiosity. (Contributed by NM, 1-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
om0x (𝐴 ·o ∅) = ∅

Proof of Theorem om0x
StepHypRef Expression
1 om0 8000 . . 3 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
21adantr 481 . 2 ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅)
3 fnom 7992 . . . 4 ·o Fn (On × On)
4 fndm 6332 . . . 4 ( ·o Fn (On × On) → dom ·o = (On × On))
53, 4ax-mp 5 . . 3 dom ·o = (On × On)
65ndmov 7195 . 2 (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅)
72, 6pm2.61i 183 1 (𝐴 ·o ∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   = wceq 1525   ∈ wcel 2083  ∅c0 4217   × cxp 5448  dom cdm 5450  Oncon0 6073   Fn wfn 6227  (class class class)co 7023   ·o comu 7958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-omul 7965 This theorem is referenced by: (None)
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