Theorem om0x 4151

Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 4149, 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.

Assertion

Ref	Expression
om0x	⊢ (A ·o ∅) = ∅

Proof of Theorem om0x

Step	Hyp	Ref	Expression
1		om0 4149	. . 3 ⊢ (A ∈ On → (A ·o ∅) = ∅)
2	1	adantr 389	. 2 ⊢ ((A ∈ On ⋀ ∅ ∈ On) → (A ·o ∅) = ∅)
3		0ex 2707	. . 3 ⊢ ∅ ∈ V
4		fnom 4142	. . . 4 ⊢ ·o Fn (On × On)
5		fndm 3583	. . . 4 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On))
6	4, 5	ax-mp 7	. . 3 ⊢ dom ·o = (On × On)
7	3, 6	ndmopr 4040	. 2 ⊢ (¬ (A ∈ On ⋀ ∅ ∈ On) → (A ·o ∅) = ∅)
8	2, 7	pm2.61i 126	1 ⊢ (A ·o ∅) = ∅

Syntax hints:   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∅c0 2277  Oncon0 2944   × cxp 3164  dom cdm 3166   Fn wfn 3173  (class class class)co 3958   ·_o comu 4124

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-omul 4129

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