Theorem oeordsuc 8552

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 6360	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	ex 415	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3	2	adantr 483	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
4		oewordri 8550	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
5	4	3adant1 1139	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
6		oecl 8494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
7	6	3adant2 1140	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
8		oecl 8494	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o 𝐶) ∈ On)
9	8	3adant1 1139	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o 𝐶) ∈ On)
10		simp1 1145	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11		omwordri 8529	. . . . . . . . . . 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 1386	. . . . . . . . . 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 8484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
15	14	3adant2 1140	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
16	15	sseq1d 3962	. . . . . . . . 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 261	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
18		ne0i 4288	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
19		on0eln0 6392	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
20	18, 19	imbitrrid 248	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
21	20	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
22		oen0 8544	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵 ↑o 𝐶))
23	22	ex 415	. . . . . . . . . . . 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 8523	. . . . . . . . . . . . . 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 610	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵 ↑o 𝐶)) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
27	26	ex 415	. . . . . . . . . . . 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 1139	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
31		oesuc 8484	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) = ((𝐵 ↑o 𝐶) ·o 𝐵))
32	31	3adant1 1139	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) = ((𝐵 ↑o 𝐶) ·o 𝐵))
33	32	eleq2d 2842	. . . . . . . . 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 261	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ (𝐵 ↑o suc 𝐶)))
35	17, 34	jcad 519	. . . . . . 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 1127	. . . . . 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 7786	. . . . . . 7 ⊢ (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38		oecl 8494	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) ∈ On)
39		oecl 8494	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) ∈ On)
40		ontr2 6383	. . . . . . . . 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 604	. . . . . . . 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 687	. . . . . . 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 602	. . . . . 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 422	. . . 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 409	. 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 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951   ⊆ wss 3899  ∅c0 4280  Oncon0 6335  suc csuc 6337  (class class class)co 7385   ·_o comu 8423   ↑_o coe 8424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-oadd 8429  df-omul 8430  df-oexp 8431

This theorem is referenced by: (None)