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Theorem om00 7700
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 2915 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2 eloni 5771 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
3 ordge1n0 7623 . . . . . . . . . 10 (Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))
42, 3syl 17 . . . . . . . . 9 (𝐴 ∈ On → (1𝑜𝐴𝐴 ≠ ∅))
54biimprd 238 . . . . . . . 8 (𝐴 ∈ On → (𝐴 ≠ ∅ → 1𝑜𝐴))
65adantr 480 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → 1𝑜𝐴))
7 on0eln0 5818 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
87adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
9 omword1 7698 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))
109ex 449 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
118, 10sylbird 250 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
126, 11anim12d 585 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (1𝑜𝐴𝐴 ⊆ (𝐴 ·𝑜 𝐵))))
13 sstr 3644 . . . . . 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 233 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → 1𝑜 ⊆ (𝐴 ·𝑜 𝐵)))
16 omcl 7661 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
17 eloni 5771 . . . . 5 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
18 ordge1n0 7623 . . . . 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 229 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) ≠ ∅))
2120necon4bd 2843 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22 oveq1 6697 . . . . . 6 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
23 om0r 7664 . . . . . 6 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
2422, 23sylan9eqr 2707 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
2524ex 449 . . . 4 (𝐵 ∈ On → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
2625adantl 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
27 oveq2 6698 . . . . . 6 (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
28 om0 7642 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
2927, 28sylan9eqr 2707 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
3029ex 449 . . . 4 (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
3130adantr 480 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
3226, 31jaod 394 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅))
3321, 32impbid 202 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wss 3607  c0 3948  Ord word 5760  Oncon0 5761  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610
This theorem is referenced by:  om00el  7701  omlimcl  7703  oeoe  7724
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