Theorem oeordsuc 8504

Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)

Assertion

Ref	Expression
oeordsuc	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))

Proof of Theorem oeordsuc

Step	Hyp	Ref	Expression
1		onelon 6327	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	ex 412	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3	2	adantr 480	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
4		oewordri 8502	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
5	4	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
6		oecl 8447	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
7	6	3adant2 1131	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
8		oecl 8447	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o 𝐶) ∈ On)
9	8	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o 𝐶) ∈ On)
10		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11		omwordri 8482	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝐶) ∈ On ∧ (𝐵 ↑o 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶) → ((𝐴 ↑o 𝐶) ·o 𝐴) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
12	7, 9, 10, 11	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶) → ((𝐴 ↑o 𝐶) ·o 𝐴) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
13	5, 12	syld 47	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐴 ↑o 𝐶) ·o 𝐴) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
14		oesuc 8437	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
15	14	3adant2 1131	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
16	15	sseq1d 3964	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ↔ ((𝐴 ↑o 𝐶) ·o 𝐴) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
17	13, 16	sylibrd 259	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
18		ne0i 4289	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
19		on0eln0 6359	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
20	18, 19	imbitrrid 246	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
22		oen0 8496	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵 ↑o 𝐶))
23	22	ex 412	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵 ↑o 𝐶)))
24	21, 23	syld 47	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ∅ ∈ (𝐵 ↑o 𝐶)))
25		omordi 8476	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ On ∧ (𝐵 ↑o 𝐶) ∈ On) ∧ ∅ ∈ (𝐵 ↑o 𝐶)) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
26	8, 25	syldanl 602	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵 ↑o 𝐶)) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
27	26	ex 412	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ↑o 𝐶) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵))))
28	27	com23 86	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (∅ ∈ (𝐵 ↑o 𝐶) → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵))))
29	24, 28	mpdd 43	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
30	29	3adant1 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
31		oesuc 8437	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) = ((𝐵 ↑o 𝐶) ·o 𝐵))
32	31	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) = ((𝐵 ↑o 𝐶) ·o 𝐵))
33	32	eleq2d 2815	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶) ↔ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
34	30, 33	sylibrd 259	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶)))
35	17, 34	jcad 512	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ∧ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶))))
36	35	3expa 1118	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ∧ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶))))
37		onsucb 7742	. . . . . . 7 ⊢ (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38		oecl 8447	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) ∈ On)
39		oecl 8447	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) ∈ On)
40		ontr2 6350	. . . . . . . . 9 ⊢ (((𝐴 ↑o suc 𝐶) ∈ On ∧ (𝐵 ↑o suc 𝐶) ∈ On) → (((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ∧ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶)) → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))
41	38, 39, 40	syl2an 596	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ suc 𝐶 ∈ On) ∧ (𝐵 ∈ On ∧ suc 𝐶 ∈ On)) → (((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ∧ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶)) → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))
42	41	anandirs 679	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc 𝐶 ∈ On) → (((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ∧ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶)) → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))
43	37, 42	sylan2b 594	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴) ∧ ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶)) → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))
44	36, 43	syld 47	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))
45	44	exp31 419	. . . 4 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))))
46	45	com4l 92	. . 3 ⊢ (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))))
47	46	imp 406	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶))))
48	3, 47	mpdd 43	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ≠ wne 2926   ⊆ wss 3900  ∅c0 4281  Oncon0 6302  suc csuc 6304  (class class class)co 7341   ·_o comu 8378   ↑_o coe 8379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-oadd 8384  df-omul 8385  df-oexp 8386

This theorem is referenced by: (None)