Theorem oen0 8533

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 7365	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2	1	eleq2d 2823	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o ∅)))
3		oveq2 7365	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
4	3	eleq2d 2823	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝑦)))
5		oveq2 7365	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
6	5	eleq2d 2823	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o suc 𝑦)))
7		oveq2 7365	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
8	7	eleq2d 2823	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝐵)))
9		0lt1o 8450	. . . . . . 7 ⊢ ∅ ∈ 1o
10		oe0 8468	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
11	9, 10	eleqtrrid 2845	. . . . . 6 ⊢ (𝐴 ∈ On → ∅ ∈ (𝐴 ↑o ∅))
12	11	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o ∅))
13		oecl 8483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
14		omordi 8513	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
15		om0 8463	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o ∅) = ∅)
16	15	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o 𝑦) ∈ On → (((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
17	16	ad2antlr 725	. . . . . . . . . . . 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 238	. . . . . . . . . . 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 8473	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
21	20	eleq2d 2823	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
22	21	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
23	19, 22	sylibrd 258	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦)))
24	23	exp31 420	. . . . . . . 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 411	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o suc 𝑦))))
28		0ellim 6380	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		eqimss2 4001	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o ∅) = 1o → 1o ⊆ (𝐴 ↑o ∅))
30	10, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1o ⊆ (𝐴 ↑o ∅))
31		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑o 𝑦) = (𝐴 ↑o ∅))
32	31	sseq2d 3976	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1o ⊆ (𝐴 ↑o 𝑦) ↔ 1o ⊆ (𝐴 ↑o ∅)))
33	32	rspcev 3581	. . . . . . . . . . . 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 5006	. . . . . . . . . . 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 715	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
38		vex 3449	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
39		oelim 8480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
40	38, 39	mpanlr1 704	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
41	40	anasss 467	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
42	41	an12s 647	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43	37, 42	sseqtrrd 3985	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴 ↑o 𝑥))
44		limelon 6381	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
45	38, 44	mpan 688	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
46		oecl 8483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
47	46	ancoms 459	. . . . . . . . . . 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 6327	. . . . . . . . . 10 ⊢ ((𝐴 ↑o 𝑥) ∈ On → Ord (𝐴 ↑o 𝑥))
50		ordgt0ge1 8439	. . . . . . . . . 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 715	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴 ↑o 𝑥) ↔ 1o ⊆ (𝐴 ↑o 𝑥)))
53	43, 52	mpbird 256	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑o 𝑥))
54	53	ex 413	. . . . . 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 7801	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵)))
57	56	expd 416	. . 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 418	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  ∪ ciun 4954  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1_oc1o 8405   ·_o comu 8410   ↑_o coe 8411

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-oexp 8418

This theorem is referenced by:  oeordi  8534  oeordsuc  8541  oeoelem  8545  oelimcl  8547  oeeui  8549  cantnflt  9608  cnfcom  9636  infxpenc  9954  infxpenc2  9958  onexoegt  41564  cantnftermord  41640  oacl2g  41650  omabs2  41651  omcl2  41652  ofoaf  41655  ofoafo  41656