Theorem oelimcl 7567

Description: The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
oelimcl	⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑𝑜 𝐵))

Proof of Theorem oelimcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3694	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
2		limelon 5705	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 7504	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
4	1, 2, 3	syl2an 493	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) ∈ On)
5		eloni 5650	. . 3 ⊢ ((𝐴 ↑𝑜 𝐵) ∈ On → Ord (𝐴 ↑𝑜 𝐵))
6	4, 5	syl 17	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ↑𝑜 𝐵))
7	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
8	2	adantl 481	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9		dif20el 7472	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
10	9	adantr 480	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 7553	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵))
12	7, 8, 10, 11	syl21anc 1317	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑𝑜 𝐵))
13		oelim2 7562	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦))
14	1, 13	sylan 487	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑𝑜 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦))
15	14	eleq2d 2673	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑𝑜 𝐵) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦)))
16		eliun 4460	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴 ↑𝑜 𝑦))
17		eldifi 3694	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ 1𝑜) → 𝑦 ∈ 𝐵)
18	7	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝐴 ∈ On)
19	8	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝐵 ∈ On)
20		simprl 790	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 5665	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑦 ∈ On)
23		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
24	18, 22, 23	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 𝑦) ∈ On)
25		eloni 5650	. . . . . . . . . . 11 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → Ord (𝐴 ↑𝑜 𝑦))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → Ord (𝐴 ↑𝑜 𝑦))
27		simprr 792	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑥 ∈ (𝐴 ↑𝑜 𝑦))
28		ordsucss 6910	. . . . . . . . . 10 ⊢ (Ord (𝐴 ↑𝑜 𝑦) → (𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ⊆ (𝐴 ↑𝑜 𝑦)))
29	26, 27, 28	sylc 63	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → suc 𝑥 ⊆ (𝐴 ↑𝑜 𝑦))
30		simpll 786	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝐴 ∈ (On ∖ 2𝑜))
31		oeordi 7554	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦 ∈ 𝐵 → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 𝐵)))
32	19, 30, 31	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → (𝑦 ∈ 𝐵 → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 𝐵)))
33	20, 32	mpd 15	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 𝐵))
34		onelon 5665	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑𝑜 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → 𝑥 ∈ On)
36		suceloni 6905	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → suc 𝑥 ∈ On)
38	4	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → (𝐴 ↑𝑜 𝐵) ∈ On)
39		ontr2 5689	. . . . . . . . . 10 ⊢ ((suc 𝑥 ∈ On ∧ (𝐴 ↑𝑜 𝐵) ∈ On) → ((suc 𝑥 ⊆ (𝐴 ↑𝑜 𝑦) ∧ (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 𝐵)) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
40	37, 38, 39	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → ((suc 𝑥 ⊆ (𝐴 ↑𝑜 𝑦) ∧ (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 𝐵)) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
41	29, 33, 40	mp2and 711	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑𝑜 𝑦))) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵))
42	41	expr 641	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
43	17, 42	sylan2 490	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1𝑜)) → (𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
44	43	rexlimdva 3013	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1𝑜)𝑥 ∈ (𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
45	16, 44	syl5bi 231	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1𝑜)(𝐴 ↑𝑜 𝑦) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
46	15, 45	sylbid 229	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑𝑜 𝐵) → suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
47	46	ralrimiv 2948	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑𝑜 𝐵)suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵))
48		dflim4 6940	. 2 ⊢ (Lim (𝐴 ↑𝑜 𝐵) ↔ (Ord (𝐴 ↑𝑜 𝐵) ∧ ∅ ∈ (𝐴 ↑𝑜 𝐵) ∧ ∀𝑥 ∈ (𝐴 ↑𝑜 𝐵)suc 𝑥 ∈ (𝐴 ↑𝑜 𝐵)))
49	6, 12, 47, 48	syl3anbrc 1239	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑𝑜 𝐵))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   ↑_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-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oaabs2  7612  omabs  7614