Theorem oelimcl 8538

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 4085	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
2		limelon 6390	. . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
3		oecl 8474	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2an 597	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) ∈ On)
5		eloni 6335	. . 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 8442	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
10	9	adantr 480	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ 𝐴)
11		oen0 8524	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))
12	7, 8, 10, 11	syl21anc 838	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∅ ∈ (𝐴 ↑o 𝐵))
13		oelim2 8533	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
14	1, 13	sylan 581	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦))
15	14	eleq2d 2823	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦)))
16		eliun 4952	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦))
17		eldifi 4085	. . . . . . 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 771	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ 𝐵)
21		onelon 6350	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
22	19, 20, 21	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑦 ∈ On)
23		oecl 8474	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
24	18, 22, 23	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → (𝐴 ↑o 𝑦) ∈ On)
25		eloni 6335	. . . . . . . . . . 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 773	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ (𝐴 ↑o 𝑦))
28		ordsucss 7770	. . . . . . . . . 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 767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝐴 ∈ (On ∖ 2o))
31		oeordi 8525	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ 𝐵 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o 𝐵)))
32	19, 30, 31	syl2anc 585	. . . . . . . . . 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 6350	. . . . . . . . . . . 12 ⊢ (((𝐴 ↑o 𝑦) ∈ On ∧ 𝑥 ∈ (𝐴 ↑o 𝑦)) → 𝑥 ∈ On)
35	24, 27, 34	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → 𝑥 ∈ On)
36		onsuc 7765	. . . . . . . . . . 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 6373	. . . . . . . . . 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 585	. . . . . . . . 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 700	. . . . . . . 8 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 ↑o 𝑦))) → suc 𝑥 ∈ (𝐴 ↑o 𝐵))
42	41	expr 456	. . . . . . 7 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
43	17, 42	sylan2 594	. . . . . 6 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ 𝑦 ∈ (𝐵 ∖ 1o)) → (𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
44	43	rexlimdva 3139	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (∃𝑦 ∈ (𝐵 ∖ 1o)𝑥 ∈ (𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
45	16, 44	biimtrid 242	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ ∪ 𝑦 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑦) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
46	15, 45	sylbid 240	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝑥 ∈ (𝐴 ↑o 𝐵) → suc 𝑥 ∈ (𝐴 ↑o 𝐵)))
47	46	ralrimiv 3129	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → ∀𝑥 ∈ (𝐴 ↑o 𝐵)suc 𝑥 ∈ (𝐴 ↑o 𝐵))
48		dflim4 7800	. 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 1345	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ∪ ciun 4948  Ord word 6324  Oncon0 6325  Lim wlim 6326  suc csuc 6327  (class class class)co 7368  1_oc1o 8400  2_oc2o 8401   ↑_o coe 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412  df-oexp 8413

This theorem is referenced by:  oaabs2  8587  omabs  8589  rp-oelim2  43665