Theorem omword 8571

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 8568	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
2		3anrot 1117	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On))
3		omcan 8570	. . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ↔ 𝐴 = 𝐵))
4	2, 3	sylanbr 594	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ↔ 𝐴 = 𝐵))
5	4	bicomd 226	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))
6	1, 5	orbi12d 932	. 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 6403	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8	7	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
9	8	adantr 486	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
10		omcl 8537	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
11		omcl 8537	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·o 𝐵) ∈ On)
12	10, 11	anim12dan 631	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On))
13	12	ancoms 464	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On))
14	13	3impa 1127	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On))
15		onsseleq 6403	. . . 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 18	. . 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 486	. 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 314	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ∅c0 4279  Oncon0 6361  (class class class)co 7418   ·_o comu 8467

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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 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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-oadd 8473  df-omul 8474

This theorem is used by:  omwordi  8572  omeulem2  8584  oeeui  8604