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Theorem nnmword 7658
Description: Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
nnmword (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem nnmword
StepHypRef Expression
1 iba 524 . . . 4 (∅ ∈ 𝐶 → (𝐵𝐴 ↔ (𝐵𝐴 ∧ ∅ ∈ 𝐶)))
2 nnmord 7657 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵𝐴 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
323com12 1266 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐵𝐴 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
41, 3sylan9bbr 736 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 ↔ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
54notbid 308 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ 𝐵𝐴 ↔ ¬ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
6 simpl1 1062 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
7 nnon 7018 . . . 4 (𝐴 ∈ ω → 𝐴 ∈ On)
86, 7syl 17 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ On)
9 simpl2 1063 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
10 nnon 7018 . . . 4 (𝐵 ∈ ω → 𝐵 ∈ On)
119, 10syl 17 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ On)
12 ontri1 5716 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
138, 11, 12syl2anc 692 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
14 simpl3 1064 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
15 nnmcl 7637 . . . . 5 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) ∈ ω)
1614, 6, 15syl2anc 692 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ω)
17 nnon 7018 . . . 4 ((𝐶 ·𝑜 𝐴) ∈ ω → (𝐶 ·𝑜 𝐴) ∈ On)
1816, 17syl 17 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ On)
19 nnmcl 7637 . . . . 5 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) ∈ ω)
2014, 9, 19syl2anc 692 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐵) ∈ ω)
21 nnon 7018 . . . 4 ((𝐶 ·𝑜 𝐵) ∈ ω → (𝐶 ·𝑜 𝐵) ∈ On)
2220, 21syl 17 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐵) ∈ On)
23 ontri1 5716 . . 3 (((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ¬ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
2418, 22, 23syl2anc 692 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ¬ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
255, 13, 243bitr4d 300 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wss 3555  c0 3891  Oncon0 5682  (class class class)co 6604  ωcom 7012   ·𝑜 comu 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510
This theorem is referenced by:  nnmcan  7659  nnmwordi  7660
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