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Theorem om00 8186
 Description: The product of two ordinal numbers is zero iff at least one of them is zero. Proposition 8.22 of [TakeutiZaring] p. 64. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
om00 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))

Proof of Theorem om00
StepHypRef Expression
1 neanior 3079 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2 eloni 6169 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
3 ordge1n0 8108 . . . . . . . . . 10 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
42, 3syl 17 . . . . . . . . 9 (𝐴 ∈ On → (1o𝐴𝐴 ≠ ∅))
54biimprd 251 . . . . . . . 8 (𝐴 ∈ On → (𝐴 ≠ ∅ → 1o𝐴))
65adantr 484 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → 1o𝐴))
7 on0eln0 6214 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
87adantl 485 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
9 omword1 8184 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))
109ex 416 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·o 𝐵)))
118, 10sylbird 263 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ → 𝐴 ⊆ (𝐴 ·o 𝐵)))
126, 11anim12d 611 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (1o𝐴𝐴 ⊆ (𝐴 ·o 𝐵))))
13 sstr 3923 . . . . . 6 ((1o𝐴𝐴 ⊆ (𝐴 ·o 𝐵)) → 1o ⊆ (𝐴 ·o 𝐵))
1412, 13syl6 35 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → 1o ⊆ (𝐴 ·o 𝐵)))
151, 14syl5bir 246 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → 1o ⊆ (𝐴 ·o 𝐵)))
16 omcl 8146 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
17 eloni 6169 . . . . 5 ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
18 ordge1n0 8108 . . . . 5 (Ord (𝐴 ·o 𝐵) → (1o ⊆ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
1916, 17, 183syl 18 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
2015, 19sylibd 242 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) ≠ ∅))
2120necon4bd 3007 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22 oveq1 7142 . . . . . 6 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
23 om0r 8149 . . . . . 6 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
2422, 23sylan9eqr 2855 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
2524ex 416 . . . 4 (𝐵 ∈ On → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
2625adantl 485 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
27 oveq2 7143 . . . . . 6 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
28 om0 8127 . . . . . 6 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2927, 28sylan9eqr 2855 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
3029ex 416 . . . 4 (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
3130adantr 484 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
3226, 31jaod 856 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅))
3321, 32impbid 215 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ⊆ wss 3881  ∅c0 4243  Ord word 6158  Oncon0 6159  (class class class)co 7135  1oc1o 8080   ·o comu 8085 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-omul 8092 This theorem is referenced by:  om00el  8187  omlimcl  8189  oeoe  8210
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