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Theorem oeword 8232
 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 8230 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ∈ (𝐶o 𝐵)))
2 oecan 8231 . . . . 5 ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
323coml 1124 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) = (𝐶o 𝐵) ↔ 𝐴 = 𝐵))
43bicomd 226 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶o 𝐴) = (𝐶o 𝐵)))
51, 4orbi12d 916 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
6 onsseleq 6215 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
763adant3 1129 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
8 eldifi 4034 . . . 4 (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9 id 22 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10 oecl 8178 . . . . . 6 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶o 𝐴) ∈ On)
11 oecl 8178 . . . . . 6 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶o 𝐵) ∈ On)
1210, 11anim12dan 621 . . . . 5 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶o 𝐴) ∈ On ∧ (𝐶o 𝐵) ∈ On))
13 onsseleq 6215 . . . . 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 599 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
16153impa 1107 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶o 𝐴) ⊆ (𝐶o 𝐵) ↔ ((𝐶o 𝐴) ∈ (𝐶o 𝐵) ∨ (𝐶o 𝐴) = (𝐶o 𝐵))))
175, 7, 163bitr4d 314 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∖ cdif 3857   ⊆ wss 3860  Oncon0 6174  (class class class)co 7156  2oc2o 8112   ↑o coe 8117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-2o 8119  df-oadd 8122  df-omul 8123  df-oexp 8124 This theorem is referenced by:  oewordi  8233
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