Theorem oen0 8550

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 7395	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2	1	eleq2d 2814	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o ∅)))
3		oveq2 7395	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
4	3	eleq2d 2814	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝑦)))
5		oveq2 7395	. . . . . 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 7395	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
8	7	eleq2d 2814	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝐵)))
9		0lt1o 8468	. . . . . . 7 ⊢ ∅ ∈ 1o
10		oe0 8486	. . . . . . 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 8501	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
14		omordi 8530	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
15		om0 8481	. . . . . . . . . . . . . 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 8491	. . . . . . . . . . . 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 6396	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		eqimss2 4006	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o ∅) = 1o → 1o ⊆ (𝐴 ↑o ∅))
30	10, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1o ⊆ (𝐴 ↑o ∅))
31		oveq2 7395	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑o 𝑦) = (𝐴 ↑o ∅))
32	31	sseq2d 3979	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1o ⊆ (𝐴 ↑o 𝑦) ↔ 1o ⊆ (𝐴 ↑o ∅)))
33	32	rspcev 3588	. . . . . . . . . . . 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 5010	. . . . . . . . . . 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 3451	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
39		oelim 8498	. . . . . . . . . . . 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 3984	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴 ↑o 𝑥))
44		limelon 6397	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
45	38, 44	mpan 690	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
46		oecl 8501	. . . . . . . . . . . 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 6342	. . . . . . . . . 10 ⊢ ((𝐴 ↑o 𝑥) ∈ On → Ord (𝐴 ↑o 𝑥))
50		ordgt0ge1 8457	. . . . . . . . . 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 7841	. . . 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 3447   ⊆ wss 3914  ∅c0 4296  ∪ ciun 4955  Ord word 6331  Oncon0 6332  Lim wlim 6333  suc csuc 6334  (class class class)co 7387  1_oc1o 8427   ·_o comu 8432   ↑_o coe 8433

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 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-omul 8439  df-oexp 8440

This theorem is referenced by:  oeordi  8551  oeordsuc  8558  oeoelem  8562  oelimcl  8564  oeeui  8566  cantnflt  9625  cnfcom  9653  infxpenc  9971  infxpenc2  9975  onexoegt  43233  cantnftermord  43309  oacl2g  43319  onmcl  43320  omabs2  43321  omcl2  43322  ofoaf  43344  ofoafo  43345