Theorem oen0 8215

Description: Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.)

Assertion

Ref	Expression
oen0	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))

Proof of Theorem oen0

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

Step	Hyp	Ref	Expression
1		oveq2 7151	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2	1	eleq2d 2836	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o ∅)))
3		oveq2 7151	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
4	3	eleq2d 2836	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝑦)))
5		oveq2 7151	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
6	5	eleq2d 2836	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o suc 𝑦)))
7		oveq2 7151	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
8	7	eleq2d 2836	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝐵)))
9		0lt1o 8132	. . . . . . 7 ⊢ ∅ ∈ 1o
10		oe0 8150	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
11	9, 10	eleqtrrid 2858	. . . . . 6 ⊢ (𝐴 ∈ On → ∅ ∈ (𝐴 ↑o ∅))
12	11	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o ∅))
13		oecl 8165	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
14		omordi 8195	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
15		om0 8145	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o ∅) = ∅)
16	15	eleq1d 2835	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o 𝑦) ∈ On → (((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
17	16	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
18	14, 17	sylibd 242	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
19	13, 18	syldanl 605	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
20		oesuc 8155	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
21	20	eleq2d 2836	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
22	21	adantr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
23	19, 22	sylibrd 262	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦)))
24	23	exp31 424	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑦 ∈ On → (∅ ∈ (𝐴 ↑o 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦)))))
25	24	com12 32	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ (𝐴 ↑o 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦)))))
26	25	com34 91	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o suc 𝑦)))))
27	26	impd 415	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o suc 𝑦))))
28		0ellim 6224	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		eqimss2 3945	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o ∅) = 1o → 1o ⊆ (𝐴 ↑o ∅))
30	10, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1o ⊆ (𝐴 ↑o ∅))
31		oveq2 7151	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑o 𝑦) = (𝐴 ↑o ∅))
32	31	sseq2d 3920	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1o ⊆ (𝐴 ↑o 𝑦) ↔ 1o ⊆ (𝐴 ↑o ∅)))
33	32	rspcev 3539	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴 ↑o ∅)) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦))
34	28, 30, 33	syl2an 599	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦))
35		ssiun 4928	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦) → 1o ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → 1o ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
37	36	adantrr 717	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
38		vex 3411	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
39		oelim 8162	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
40	38, 39	mpanlr1 706	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
41	40	anasss 471	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
42	41	an12s 649	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43	37, 42	sseqtrrd 3929	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴 ↑o 𝑥))
44		limelon 6225	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
45	38, 44	mpan 690	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
46		oecl 8165	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
47	46	ancoms 463	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
48	45, 47	sylan 584	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
49		eloni 6172	. . . . . . . . . 10 ⊢ ((𝐴 ↑o 𝑥) ∈ On → Ord (𝐴 ↑o 𝑥))
50		ordgt0ge1 8125	. . . . . . . . . 10 ⊢ (Ord (𝐴 ↑o 𝑥) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆ (𝐴 ↑o 𝑥)))
51	48, 49, 50	3syl 18	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆ (𝐴 ↑o 𝑥)))
52	51	adantrr 717	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆ (𝐴 ↑o 𝑥)))
53	43, 52	mpbird 260	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑o 𝑥))
54	53	ex 417	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝑥)))
55	54	a1dd 50	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 ∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o 𝑥))))
56	2, 4, 6, 8, 12, 27, 55	tfinds3 7571	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵)))
57	56	expd 420	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o 𝐵))))
58	57	com12 32	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o 𝐵))))
59	58	imp31 422	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3068  ∃wrex 3069  Vcvv 3407   ⊆ wss 3854  ∅c0 4221  ∪ ciun 4876  Ord word 6161  Oncon0 6162  Lim wlim 6163  suc csuc 6164  (class class class)co 7143  1_oc1o 8098   ·_o comu 8103   ↑_o coe 8104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-omul 8110  df-oexp 8111

This theorem is referenced by:  oeordi  8216  oeordsuc  8223  oeoelem  8227  oelimcl  8229  oeeui  8231  cantnflt  9153  cnfcom  9181  infxpenc  9463  infxpenc2  9467