Theorem oeword 7845

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 7843	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
2		oecan 7844	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵))
3	2	3coml 1148	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵))
4	3	bicomd 214	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 = 𝐵 ↔ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))
5	1, 4	orbi12d 931	. 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 5919	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	6	3adant3 1153	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		eldifi 3890	. . . 4 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
9		id 22	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10		oecl 7792	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
11		oecl 7792	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑𝑜 𝐵) ∈ On)
12	10, 11	anim12dan 606	. . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On))
13		onsseleq 5919	. . . . 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 585	. . 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 1127	. 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 301	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 383   ∨ wo 863   ∧ w3a 1098   = wceq 1634   ∈ wcel 2148   ∖ cdif 3726   ⊆ wss 3729  Oncon0 5877  (class class class)co 6812  2_𝑜c2o 7728   ↑_𝑜 coe 7733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117

This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-2o 7735  df-oadd 7738  df-omul 7739  df-oexp 7740

This theorem is referenced by:  oewordi  7846