Theorem omword 8495

Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Assertion

Ref	Expression
omword	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Proof of Theorem omword

Step	Hyp	Ref	Expression
1		omord2 8492	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
2		3anrot 1105	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On))
3		omcan 8494	. . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ↔ 𝐴 = 𝐵))
4	2, 3	sylanbr 588	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ↔ 𝐴 = 𝐵))
5	4	bicomd 224	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))
6	1, 5	orbi12d 924	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))))
7		onsseleq 6351	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8	7	3adant3 1138	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
9	8	adantr 481	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
10		omcl 8461	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
11		omcl 8461	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·o 𝐵) ∈ On)
12	10, 11	anim12dan 625	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On))
13	12	ancoms 459	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On))
14	13	3impa 1115	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On))
15		onsseleq 6351	. . . 4 ⊢ (((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))))
16	14, 15	syl 17	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))))
17	16	adantr 481	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))))
18	6, 9, 17	3bitr4d 312	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ∅c0 4261  Oncon0 6310  (class class class)co 7356   ·_o comu 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399  df-omul 8400

This theorem is referenced by:  omwordi  8496  omeulem2  8508  oeeui  8528