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Theorem om00 7601
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 2888 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2 eloni 5695 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
3 ordge1n0 7524 . . . . . . . . . 10 (Ord 𝐴 → (1𝑜𝐴𝐴 ≠ ∅))
42, 3syl 17 . . . . . . . . 9 (𝐴 ∈ On → (1𝑜𝐴𝐴 ≠ ∅))
54biimprd 238 . . . . . . . 8 (𝐴 ∈ On → (𝐴 ≠ ∅ → 1𝑜𝐴))
65adantr 481 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → 1𝑜𝐴))
7 on0eln0 5742 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
87adantl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
9 omword1 7599 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))
109ex 450 . . . . . . . 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 3596 . . . . . 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 7562 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
17 eloni 5695 . . . . 5 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
18 ordge1n0 7524 . . . . 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 2816 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22 oveq1 6612 . . . . . 6 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
23 om0r 7565 . . . . . 6 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
2422, 23sylan9eqr 2682 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
2524ex 450 . . . 4 (𝐵 ∈ On → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
2625adantl 482 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
27 oveq2 6613 . . . . . 6 (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
28 om0 7543 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
2927, 28sylan9eqr 2682 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
3029ex 450 . . . 4 (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
3130adantr 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
3226, 31jaod 395 . 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 383  wa 384   = wceq 1480  wcel 1992  wne 2796  wss 3560  c0 3896  Ord word 5684  Oncon0 5685  (class class class)co 6605  1𝑜c1o 7499   ·𝑜 comu 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511
This theorem is referenced by:  om00el  7602  omlimcl  7604  oeoe  7625
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