Theorem oeordsuc 8387

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 6276	. . . 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 8385	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
5	4	3adant1 1128	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶)))
6		oecl 8329	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
7	6	3adant2 1129	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
8		oecl 8329	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o 𝐶) ∈ On)
9	8	3adant1 1128	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o 𝐶) ∈ On)
10		simp1 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11		omwordri 8365	. . . . . . . . . . 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 1369	. . . . . . . . . 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 8319	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
15	14	3adant2 1129	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) = ((𝐴 ↑o 𝐶) ·o 𝐴))
16	15	sseq1d 3948	. . . . . . . . 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 258	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ⊆ ((𝐵 ↑o 𝐶) ·o 𝐴)))
18		ne0i 4265	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
19		on0eln0 6306	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
20	18, 19	syl5ibr 245	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
22		oen0 8379	. . . . . . . . . . . . 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 8359	. . . . . . . . . . . . . 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 601	. . . . . . . . . . . . 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 1128	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑o 𝐶) ·o 𝐴) ∈ ((𝐵 ↑o 𝐶) ·o 𝐵)))
31		oesuc 8319	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) = ((𝐵 ↑o 𝐶) ·o 𝐵))
32	31	3adant1 1128	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) = ((𝐵 ↑o 𝐶) ·o 𝐵))
33	32	eleq2d 2824	. . . . . . . . 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 258	. . . . . . . 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 1116	. . . . . 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		sucelon 7639	. . . . . . 7 ⊢ (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38		oecl 8329	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑o suc 𝐶) ∈ On)
39		oecl 8329	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵 ↑o suc 𝐶) ∈ On)
40		ontr2 6298	. . . . . . . . 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 595	. . . . . . . 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 675	. . . . . . 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 593	. . . . . 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 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ⊆ wss 3883  ∅c0 4253  Oncon0 6251  suc csuc 6253  (class class class)co 7255   ·_o comu 8265   ↑_o coe 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-oexp 8273

This theorem is referenced by: (None)