Theorem om0x 8565

Description: Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. Unlike om0 8563, 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

Step	Hyp	Ref	Expression
1		om0 8563	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2	1	adantr 486	. 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅)
3		fnom 8555	. . . 4 ⊢ ·o Fn (On × On)
4	3	fndmi 6673	. . 3 ⊢ dom ·o = (On × On)
5	4	ndmov 7637	. 2 ⊢ (¬ (𝐴 ∈ On ∧ ∅ ∈ On) → (𝐴 ·o ∅) = ∅)
6	2, 5	pm2.61i 184	1 ⊢ (𝐴 ·o ∅) = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∅c0 4281   × cxp 5656  Oncon0 6388  (class class class)co 7451   ·_o comu 8512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-omul 8519

This theorem is used by: (None)