Theorem oen0 8504

Description: Ordinal exponentiation with a nonzero base 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 7357	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2	1	eleq2d 2814	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o ∅)))
3		oveq2 7357	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
4	3	eleq2d 2814	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝑦)))
5		oveq2 7357	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
6	5	eleq2d 2814	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o suc 𝑦)))
7		oveq2 7357	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
8	7	eleq2d 2814	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝐵)))
9		0lt1o 8422	. . . . . . 7 ⊢ ∅ ∈ 1o
10		oe0 8440	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
11	9, 10	eleqtrrid 2835	. . . . . 6 ⊢ (𝐴 ∈ On → ∅ ∈ (𝐴 ↑o ∅))
12	11	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o ∅))
13		oecl 8455	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
14		omordi 8484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
15		om0 8435	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o ∅) = ∅)
16	15	eleq1d 2813	. . . . . . . . . . . . 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 239	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
19	13, 18	syldanl 602	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
20		oesuc 8445	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
21	20	eleq2d 2814	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
22	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
23	19, 22	sylibrd 259	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦)))
24	23	exp31 419	. . . . . . . 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 410	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o suc 𝑦))))
28		0ellim 6371	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		eqimss2 3995	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o ∅) = 1o → 1o ⊆ (𝐴 ↑o ∅))
30	10, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1o ⊆ (𝐴 ↑o ∅))
31		oveq2 7357	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑o 𝑦) = (𝐴 ↑o ∅))
32	31	sseq2d 3968	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1o ⊆ (𝐴 ↑o 𝑦) ↔ 1o ⊆ (𝐴 ↑o ∅)))
33	32	rspcev 3577	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴 ↑o ∅)) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦))
34	28, 30, 33	syl2an 596	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦))
35		ssiun 4995	. . . . . . . . . . 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 3440	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
39		oelim 8452	. . . . . . . . . . . 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 466	. . . . . . . . . 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 3973	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴 ↑o 𝑥))
44		limelon 6372	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
45	38, 44	mpan 690	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
46		oecl 8455	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
47	46	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
48	45, 47	sylan 580	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
49		eloni 6317	. . . . . . . . . 10 ⊢ ((𝐴 ↑o 𝑥) ∈ On → Ord (𝐴 ↑o 𝑥))
50		ordgt0ge1 8411	. . . . . . . . . 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 257	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑o 𝑥))
54	53	ex 412	. . . . . 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 7798	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵)))
57	56	expd 415	. . 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 417	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903  ∅c0 4284  ∪ ciun 4941  Ord word 6306  Oncon0 6307  Lim wlim 6308  suc csuc 6309  (class class class)co 7349  1_oc1o 8381   ·_o comu 8386   ↑_o coe 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-omul 8393  df-oexp 8394

This theorem is referenced by:  oeordi  8505  oeordsuc  8512  oeoelem  8516  oelimcl  8518  oeeui  8520  cantnflt  9568  cnfcom  9596  infxpenc  9912  infxpenc2  9916  onexoegt  43217  cantnftermord  43293  oacl2g  43303  onmcl  43304  omabs2  43305  omcl2  43306  ofoaf  43328  ofoafo  43329