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Theorem omord 7512
Description: Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)
Assertion
Ref Expression
omord ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omord
StepHypRef Expression
1 omord2 7511 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
21ex 448 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
32pm5.32rd 669 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶)))
4 simpl 471 . . 3 (((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))
5 ne0i 3879 . . . . . . . 8 ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐶 ·𝑜 𝐵) ≠ ∅)
6 om0r 7483 . . . . . . . . . 10 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
7 oveq1 6534 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
87eqeq1d 2611 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·𝑜 𝐵) = ∅ ↔ (∅ ·𝑜 𝐵) = ∅))
96, 8syl5ibrcom 235 . . . . . . . . 9 (𝐵 ∈ On → (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = ∅))
109necon3d 2802 . . . . . . . 8 (𝐵 ∈ On → ((𝐶 ·𝑜 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
115, 10syl5 33 . . . . . . 7 (𝐵 ∈ On → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
1211adantr 479 . . . . . 6 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
13 on0eln0 5683 . . . . . . 7 (𝐶 ∈ On → (∅ ∈ 𝐶𝐶 ≠ ∅))
1413adantl 480 . . . . . 6 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶𝐶 ≠ ∅))
1512, 14sylibrd 247 . . . . 5 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
16153adant1 1071 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
1716ancld 573 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶)))
184, 17impbid2 214 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
193, 18bitrd 266 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  c0 3873  Oncon0 5626  (class class class)co 6527   ·𝑜 comu 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-oadd 7428  df-omul 7429
This theorem is referenced by:  omlimcl  7522  oneo  7525
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