Theorem oeword 7715

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 7713	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
2		oecan 7714	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵))
3	2	3coml 1292	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵))
4	3	bicomd 213	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 = 𝐵 ↔ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))
5	1, 4	orbi12d 746	. 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 5803	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	6	3adant3 1101	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		eldifi 3765	. . . 4 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
9		id 22	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10		oecl 7662	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
11		oecl 7662	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑𝑜 𝐵) ∈ On)
12	10, 11	anim12dan 900	. . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On))
13		onsseleq 5803	. . . . 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 494	. . 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 1278	. 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 300	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ∖ cdif 3604   ⊆ wss 3607  Oncon0 5761  (class class class)co 6690  2_𝑜c2o 7599   ↑_𝑜 coe 7604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611

This theorem is referenced by:  oewordi  7716