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Theorem om00 8613
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 3035 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2 eloni 6394 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
3 ordge1n0 8532 . . . . . . . . . 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 6440 . . . . . . . . 9 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
87adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐵 ≠ ∅))
9 omword1 8611 . . . . . . . . 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 3992 . . . . . 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 8574 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
17 eloni 6394 . . . . 5 ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
18 ordge1n0 8532 . . . . 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 2960 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22 oveq1 7438 . . . . . 6 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
23 om0r 8577 . . . . . 6 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
2422, 23sylan9eqr 2799 . . . . 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 7439 . . . . . 6 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
28 om0 8555 . . . . . 6 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
2927, 28sylan9eqr 2799 . . . . 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 860 . 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 848   = wceq 1540  wcel 2108  wne 2940  wss 3951  c0 4333  Ord word 6383  Oncon0 6384  (class class class)co 7431  1oc1o 8499   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511
This theorem is referenced by:  om00el  8614  omlimcl  8616  oeoe  8637
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