Theorem om00 7655

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 2886	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅))
2		eloni 5733	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordge1n0 7578	. . . . . . . . . 10 ⊢ (Ord 𝐴 → (1𝑜 ⊆ 𝐴 ↔ 𝐴 ≠ ∅))
4	2, 3	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (1𝑜 ⊆ 𝐴 ↔ 𝐴 ≠ ∅))
5	4	biimprd 238	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝐴 ≠ ∅ → 1𝑜 ⊆ 𝐴))
6	5	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → 1𝑜 ⊆ 𝐴))
7		on0eln0 5780	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
8	7	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
9		omword1 7653	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))
10	9	ex 450	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
11	8, 10	sylbird 250	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ → 𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
12	6, 11	anim12d 586	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (1𝑜 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ·𝑜 𝐵))))
13		sstr 3611	. . . . . 6 ⊢ ((1𝑜 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐴 ·𝑜 𝐵)) → 1𝑜 ⊆ (𝐴 ·𝑜 𝐵))
14	12, 13	syl6 35	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → 1𝑜 ⊆ (𝐴 ·𝑜 𝐵)))
15	1, 14	syl5bir 233	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → 1𝑜 ⊆ (𝐴 ·𝑜 𝐵)))
16		omcl 7616	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
17		eloni 5733	. . . . 5 ⊢ ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
18		ordge1n0 7578	. . . . 5 ⊢ (Ord (𝐴 ·𝑜 𝐵) → (1𝑜 ⊆ (𝐴 ·𝑜 𝐵) ↔ (𝐴 ·𝑜 𝐵) ≠ ∅))
19	16, 17, 18	3syl 18	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ⊆ (𝐴 ·𝑜 𝐵) ↔ (𝐴 ·𝑜 𝐵) ≠ ∅))
20	15, 19	sylibd 229	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) ≠ ∅))
21	20	necon4bd 2814	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
22		oveq1 6657	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
23		om0r 7619	. . . . . 6 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
24	22, 23	sylan9eqr 2678	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
25	24	ex 450	. . . 4 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
26	25	adantl 482	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
27		oveq2 6658	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
28		om0 7597	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
29	27, 28	sylan9eqr 2678	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
30	29	ex 450	. . . 4 ⊢ (𝐴 ∈ On → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
31	30	adantr 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
32	26, 31	jaod 395	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅))
33	21, 32	impbid 202	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ⊆ wss 3574  ∅c0 3915  Ord word 5722  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565

This theorem is referenced by:  om00el  7656  omlimcl  7658  oeoe  7679