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Theorem oeword 7938
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 7936 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
2 oecan 7937 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
323coml 1163 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
43bicomd 215 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶o 𝐴) = (𝐶o 𝐵)))
51, 4orbi12d 949 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
6 onsseleq 6005 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
763adant3 1168 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
8 eldifi 3960 . . . 4 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9 id 22 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10 oecl 7885 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
11 oecl 7885 . . . . . 6 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1210, 11anim12dan 614 . . . . 5 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On))
13 onsseleq 6005 . . . . 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 592 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
16153impa 1142 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
175, 7, 163bitr4d 303 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 880  w3a 1113   = wceq 1658  wcel 2166  cdif 3796  wss 3799  Oncon0 5964  (class class class)co 6906  2oc2o 7821  o coe 7826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-2o 7828  df-oadd 7831  df-omul 7832  df-oexp 7833
This theorem is referenced by:  oewordi  7939
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