Theorem om00 8590

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

Step	Hyp	Ref	Expression
1		neanior 3048	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2		eloni 6375	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordge1n0 8509	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (1o ⊆ 𝐴 ↔ 𝐴 ≠ ∅))
4	2, 3	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (1o ⊆ 𝐴 ↔ 𝐴 ≠ ∅))
5	4	biimprd 251	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ≠ ∅ → 1o ⊆ 𝐴))
6	5	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → 1o ⊆ 𝐴))
7		on0eln0 6423	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
8	7	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
9		omword1 8588	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))
10	9	ex 418	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵)))
11	8, 10	sylbird 263	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ → 𝐴 ⊆ (𝐴 ·o 𝐵)))
12	6, 11	anim12d 621	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (1o ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ·o 𝐵))))
13		sstr 3917	. . . . . 6 ⊢ ((1o ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ·o 𝐵)) → 1o ⊆ (𝐴 ·o 𝐵))
14	12, 13	syl6 36	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → 1o ⊆ (𝐴 ·o 𝐵)))
15	1, 14	biimtrrid 246	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → 1o ⊆ (𝐴 ·o 𝐵)))
16		omcl 8551	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
17		eloni 6375	. . . . 5 ⊢ ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
18		ordge1n0 8509	. . . . 5 ⊢ (Ord (𝐴 ·o 𝐵) → (1o ⊆ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
19	16, 17, 18	3syl 19	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
20	15, 19	sylibd 242	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) ≠ ∅))
21	20	necon4bd 2975	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22		oveq1 7431	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
23		om0r 8554	. . . . . 6 ⊢ (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
24	22, 23	sylan9eqr 2817	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
25	24	ex 418	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
26	25	adantl 487	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
27		oveq2 7432	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
28		om0 8532	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
29	27, 28	sylan9eqr 2817	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
30	29	ex 418	. . . 4 ⊢ (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
31	30	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
32	26, 31	jaod 873	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅))
33	21, 32	impbid 215	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3896  ∅c0 4277  Ord word 6364  Oncon0 6365  (class class class)co 7424  1_oc1o 8476   ·_o comu 8481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-oadd 8487  df-omul 8488

This theorem is used by:  om00el  8591  omlimcl  8593  oeoe  8615