MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeword Structured version   Visualization version   GIF version

Theorem oeword 8209
Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeword ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))

Proof of Theorem oeword
StepHypRef Expression
1 oeord 8207 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
2 oecan 8208 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
323coml 1121 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
43bicomd 224 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶o 𝐴) = (𝐶o 𝐵)))
51, 4orbi12d 914 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
6 onsseleq 6229 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
763adant3 1126 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
8 eldifi 4106 . . . 4 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9 id 22 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10 oecl 8156 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
11 oecl 8156 . . . . . 6 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1210, 11anim12dan 618 . . . . 5 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On))
13 onsseleq 6229 . . . . 5 (((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
1412, 13syl 17 . . . 4 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
158, 9, 14syl2anr 596 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
16153impa 1104 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
175, 7, 163bitr4d 312 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  cdif 3936  wss 3939  Oncon0 6188  (class class class)co 7151  2oc2o 8090  o coe 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101  df-oexp 8102
This theorem is referenced by:  oewordi  8210
  Copyright terms: Public domain W3C validator