Theorem oen0 7553

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 6557	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
2	1	eleq2d 2673	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 ∅)))
3		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
4	3	eleq2d 2673	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 𝑦)))
5		oveq2 6557	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
6	5	eleq2d 2673	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))
7		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵))
8	7	eleq2d 2673	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 𝐵)))
9		0lt1o 7471	. . . . . . 7 ⊢ ∅ ∈ 1𝑜
10		oe0 7489	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
11	9, 10	syl5eleqr 2695	. . . . . 6 ⊢ (𝐴 ∈ On → ∅ ∈ (𝐴 ↑𝑜 ∅))
12	11	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 ∅))
13		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
14		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
15	13, 14	jca 553	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On))
16		omordi 7533	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
17		om0 7484	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → ((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) = ∅)
18	17	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → (((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
19	18	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
20	16, 19	sylibd 228	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
21	15, 20	sylan 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
22		oesuc 7494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
23	22	eleq2d 2673	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴 ↑𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
24	23	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ (𝐴 ↑𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
25	21, 24	sylibrd 248	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))
26	25	exp31 628	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑦 ∈ On → (∅ ∈ (𝐴 ↑𝑜 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))))
27	26	com12 32	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ (𝐴 ↑𝑜 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))))
28	27	com34 89	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (∅ ∈ (𝐴 ↑𝑜 𝑦) → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))))
29	28	impd 446	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴 ↑𝑜 𝑦) → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦))))
30		0ellim 5704	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
31		eqimss2 3621	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑𝑜 ∅) = 1𝑜 → 1𝑜 ⊆ (𝐴 ↑𝑜 ∅))
32	10, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1𝑜 ⊆ (𝐴 ↑𝑜 ∅))
33		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑𝑜 𝑦) = (𝐴 ↑𝑜 ∅))
34	33	sseq2d 3596	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1𝑜 ⊆ (𝐴 ↑𝑜 𝑦) ↔ 1𝑜 ⊆ (𝐴 ↑𝑜 ∅)))
35	34	rspcev 3282	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑥 ∧ 1𝑜 ⊆ (𝐴 ↑𝑜 ∅)) → ∃𝑦 ∈ 𝑥 1𝑜 ⊆ (𝐴 ↑𝑜 𝑦))
36	30, 32, 35	syl2an 493	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝑥 1𝑜 ⊆ (𝐴 ↑𝑜 𝑦))
37		ssiun 4498	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑥 1𝑜 ⊆ (𝐴 ↑𝑜 𝑦) → 1𝑜 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
38	36, 37	syl 17	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → 1𝑜 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
39	38	adantrr 749	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
40		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
41		oelim 7501	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
42	40, 41	mpanlr1 718	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
43	42	anasss 677	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
44	43	an12s 839	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
45	39, 44	sseqtr4d 3605	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 ⊆ (𝐴 ↑𝑜 𝑥))
46		limelon 5705	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
47	40, 46	mpan 702	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
48		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑𝑜 𝑥) ∈ On)
49	48	ancoms 468	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 𝑥) ∈ On)
50	47, 49	sylan 487	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 𝑥) ∈ On)
51		eloni 5650	. . . . . . . . . 10 ⊢ ((𝐴 ↑𝑜 𝑥) ∈ On → Ord (𝐴 ↑𝑜 𝑥))
52		ordgt0ge1 7464	. . . . . . . . . 10 ⊢ (Ord (𝐴 ↑𝑜 𝑥) → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴 ↑𝑜 𝑥)))
53	50, 51, 52	3syl 18	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴 ↑𝑜 𝑥)))
54	53	adantrr 749	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴 ↑𝑜 𝑥)))
55	45, 54	mpbird 246	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑𝑜 𝑥))
56	55	ex 449	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝑥)))
57	56	a1dd 48	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 ∅ ∈ (𝐴 ↑𝑜 𝑦) → ∅ ∈ (𝐴 ↑𝑜 𝑥))))
58	2, 4, 6, 8, 12, 29, 57	tfinds3 6956	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵)))
59	58	expd 451	. . 3 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑𝑜 𝐵))))
60	59	com12 32	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑𝑜 𝐵))))
61	60	imp31 447	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oeordi  7554  oeordsuc  7561  oeoelem  7565  oelimcl  7567  oeeui  7569  cantnflt  8452  cnfcom  8480  infxpenc  8724  infxpenc2  8728