Theorem oeordsuc 7561

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 5665	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ On)
2	1	ex 449	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3	2	adantr 480	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
4		oewordri 7559	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))
5	4	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ⊆ (𝐵 ↑𝑜 𝐶)))
6		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
7	6	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
8		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑𝑜 𝐶) ∈ On)
9	8	3adant1 1072	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑𝑜 𝐶) ∈ On)
10		simp1 1054	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11		omwordri 7539	. . . . . . . . . . 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 1318	. . . . . . . . . 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 46	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴)))
14		oesuc 7494	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 suc 𝐶) = ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
15	14	3adant2 1073	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 suc 𝐶) = ((𝐴 ↑𝑜 𝐶) ·𝑜 𝐴))
16	15	sseq1d 3595	. . . . . . . . 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 248	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 suc 𝐶) ⊆ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴)))
18		ne0i 3880	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ ∅)
19		on0eln0 5697	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
20	18, 19	syl5ibr 235	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ∅ ∈ 𝐵))
22		oen0 7553	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵 ↑𝑜 𝐶))
23	22	ex 449	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵 ↑𝑜 𝐶)))
24	21, 23	syld 46	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ∅ ∈ (𝐵 ↑𝑜 𝐶)))
25		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
26	25, 8	jca 553	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ On ∧ (𝐵 ↑𝑜 𝐶) ∈ On))
27		omordi 7533	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ On ∧ (𝐵 ↑𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐵 ↑𝑜 𝐶)) → (𝐴 ∈ 𝐵 → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵)))
28	26, 27	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵 ↑𝑜 𝐶)) → (𝐴 ∈ 𝐵 → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵)))
29	28	ex 449	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵 ↑𝑜 𝐶) → (𝐴 ∈ 𝐵 → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵))))
30	29	com23 84	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (∅ ∈ (𝐵 ↑𝑜 𝐶) → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵))))
31	24, 30	mpdd 42	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵)))
32	31	3adant1 1072	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵)))
33		oesuc 7494	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑𝑜 suc 𝐶) = ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵))
34	33	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ↑𝑜 suc 𝐶) = ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐵))
35	34	eleq2d 2673	. . . . . . . . 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)))
36	32, 35	sylibrd 248	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → ((𝐵 ↑𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵 ↑𝑜 suc 𝐶)))
37	17, 36	jcad 554	. . . . . . 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))))
38	37	3expa 1257	. . . . . 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))))
39		sucelon 6909	. . . . . . 7 ⊢ (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
40		oecl 7504	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴 ↑𝑜 suc 𝐶) ∈ On)
41		oecl 7504	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵 ↑𝑜 suc 𝐶) ∈ On)
42		ontr2 5689	. . . . . . . . 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)))
43	40, 41, 42	syl2an 493	. . . . . . . 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)))
44	43	anandirs 870	. . . . . . 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)))
45	39, 44	sylan2b 491	. . . . . 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)))
46	38, 45	syld 46	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 suc 𝐶) ∈ (𝐵 ↑𝑜 suc 𝐶)))
47	46	exp31 628	. . . 4 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 suc 𝐶) ∈ (𝐵 ↑𝑜 suc 𝐶)))))
48	47	com4l 90	. . 3 ⊢ (𝐵 ∈ On → (𝐶 ∈ On → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐴 ↑𝑜 suc 𝐶) ∈ (𝐵 ↑𝑜 suc 𝐶)))))
49	48	imp 444	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝐴 ↑𝑜 suc 𝐶) ∈ (𝐵 ↑𝑜 suc 𝐶))))
50	3, 49	mpdd 42	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑𝑜 suc 𝐶) ∈ (𝐵 ↑𝑜 suc 𝐶)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by: (None)