Theorem oelimcl 8570

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 4084	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2		limelon 6411	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 8506	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2an 605	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) ∈ On)
5		eloni 6356	. . 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 484	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐴 ∈ On)
8	2	adantl 485	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → 𝐵 ∈ On)
9		dif20el 8474	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
10	9	adantr 484	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 8556	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
12	7, 8, 10, 11	syl21anc 848	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑o 𝐵))
13		oelim2 8565	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
14	1, 13	sylan 589	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
15	14	eleq2d 2848	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦)))
16		eliun 4953	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦))
17		eldifi 4084	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ 1o) → 𝑦 ∈ 𝐵)
18	7	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ On)
19	8	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐵 ∈ On)
20		simprl 780	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 6371	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ On)
23		oecl 8506	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
24	18, 22, 23	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝑦) ∈ On)
25		eloni 6356	. . . . . . . . . . 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 782	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ (𝐴 ↑o 𝑦))
28		ordsucss 7798	. . . . . . . . . 10 ⊢ (Ord (𝐴 ↑o 𝑦) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ⊆ (𝐴 ↑o 𝑦)))
29	26, 27, 28	sylc 65	. . . . . . . . 9 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ⊆ (𝐴 ↑o 𝑦))
30		simpll 776	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31		oeordi 8557	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)))
32	19, 30, 31	syl2anc 593	. . . . . . . . . 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 6371	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑o 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ On)
36		onsuc 7793	. . . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝐵) ∈ On)
39		ontr2 6394	. . . . . . . . . 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 593	. . . . . . . . 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 709	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ∈ (𝐴 ↑o 𝐵))
42	41	expr 460	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
43	17, 42	sylan2 602	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
44	43	rexlimdva 3163	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
45	16, 44	biimtrid 244	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
46	15, 45	sylbid 242	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
47	46	ralrimiv 3153	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑o 𝐵)suc 𝑥 ∈ (𝐴 ↑o 𝐵))
48		dflim4 7828	. 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 1357	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  ∪ ciun 4949  Ord word 6345  Oncon0 6346  Lim wlim 6347  suc csuc 6348  (class class class)co 7396  1_oc1o 8430  2_oc2o 8431   ↑_o coe 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443

This theorem is referenced by:  oaabs2  8619  omabs  8621  rp-oelim2  43882