Theorem oeword 8520

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 8518	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2		oecan 8519	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵) ↔ 𝐴 = 𝐵))
3	2	3coml 1128	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵) ↔ 𝐴 = 𝐵))
4	3	bicomd 223	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵)))
5	1, 4	orbi12d 919	. 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 6359	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	6	3adant3 1133	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		eldifi 4084	. . . 4 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9		id 22	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10		oecl 8466	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
11		oecl 8466	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑o 𝐵) ∈ On)
12	10, 11	anim12dan 620	. . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑o 𝐴) ∈ On ∧ (𝐶 ↑o 𝐵) ∈ On))
13		onsseleq 6359	. . . . 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 598	. . 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 1110	. 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 311	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3899   ⊆ wss 3902  Oncon0 6318  (class class class)co 7360  2_oc2o 8393   ↑_o coe 8398

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-oexp 8405

This theorem is referenced by:  oewordi  8521  cantnftermord  43629  omabs2  43641