Theorem om00 8562

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 6367	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordge1n0 8481	. . . . . . . . . 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 6415	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
8	7	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
9		omword1 8560	. . . . . . . . 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 3939	. . . . . 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 8523	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
17		eloni 6367	. . . . 5 ⊢ ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
18		ordge1n0 8481	. . . . 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 7420	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
23		om0r 8526	. . . . . 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 7421	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
28		om0 8504	. . . . . 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 1570   ∈ wcel 2145   ≠ wne 2955   ⊆ wss 3899  ∅c0 4279  Ord word 6356  Oncon0 6357  (class class class)co 7413  1_oc1o 8448   ·_o comu 8453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-omul 8460

This theorem is used by:  om00el  8563  omlimcl  8565  oeoe  8587