Theorem oeword 8383

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)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Proof of Theorem oeword

Step	Hyp	Ref	Expression
1		oeord 8381	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2		oecan 8382	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵) ↔ 𝐴 = 𝐵))
3	2	3coml 1125	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵) ↔ 𝐴 = 𝐵))
4	3	bicomd 222	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵)))
5	1, 4	orbi12d 915	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) ∨ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵))))
6		onsseleq 6292	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	6	3adant3 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		eldifi 4057	. . . 4 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9		id 22	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10		oecl 8329	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
11		oecl 8329	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑o 𝐵) ∈ On)
12	10, 11	anim12dan 618	. . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑o 𝐴) ∈ On ∧ (𝐶 ↑o 𝐵) ∈ On))
13		onsseleq 6292	. . . . 5 ⊢ (((𝐶 ↑o 𝐴) ∈ On ∧ (𝐶 ↑o 𝐵) ∈ On) → ((𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵) ↔ ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) ∨ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵))))
14	12, 13	syl 17	. . . 4 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵) ↔ ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) ∨ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵))))
15	8, 9, 14	syl2anr 596	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵) ↔ ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) ∨ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵))))
16	15	3impa 1108	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵) ↔ ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) ∨ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵))))
17	5, 7, 16	3bitr4d 310	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ⊆ wss 3883  Oncon0 6251  (class class class)co 7255  2_oc2o 8261   ↑_o coe 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-oexp 8273

This theorem is referenced by:  oewordi  8384