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Theorem om00 8611
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 3032 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2 eloni 6395 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
3 ordge1n0 8530 . . . . . . . . . 10 (Ord 𝐴 → (1o𝐴𝐴 ≠ ∅))
42, 3syl 17 . . . . . . . . 9 (𝐴 ∈ On → (1o𝐴𝐴 ≠ ∅))
54biimprd 248 . . . . . . . 8 (𝐴 ∈ On → (𝐴 ≠ ∅ → 1o𝐴))
65adantr 480 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → 1o𝐴))
7 on0eln0 6441 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
87adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
9 omword1 8609 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))
109ex 412 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·o 𝐵)))
118, 10sylbird 260 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ → 𝐴 ⊆ (𝐴 ·o 𝐵)))
126, 11anim12d 609 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (1o𝐴𝐴 ⊆ (𝐴 ·o 𝐵))))
13 sstr 4003 . . . . . 6 ((1o𝐴𝐴 ⊆ (𝐴 ·o 𝐵)) → 1o ⊆ (𝐴 ·o 𝐵))
1412, 13syl6 35 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → 1o ⊆ (𝐴 ·o 𝐵)))
151, 14biimtrrid 243 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → 1o ⊆ (𝐴 ·o 𝐵)))
16 omcl 8572 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
17 eloni 6395 . . . . 5 ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
18 ordge1n0 8530 . . . . 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 239 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) ≠ ∅))
2120necon4bd 2957 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22 oveq1 7437 . . . . . 6 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
23 om0r 8575 . . . . . 6 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
2422, 23sylan9eqr 2796 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
2524ex 412 . . . 4 (𝐵 ∈ On → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
2625adantl 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
27 oveq2 7438 . . . . . 6 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
28 om0 8553 . . . . . 6 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2927, 28sylan9eqr 2796 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
3029ex 412 . . . 4 (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
3130adantr 480 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
3226, 31jaod 859 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅))
3321, 32impbid 212 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  wss 3962  c0 4338  Ord word 6384  Oncon0 6385  (class class class)co 7430  1oc1o 8497   ·o comu 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509
This theorem is referenced by:  om00el  8612  omlimcl  8614  oeoe  8635
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