Theorem oen0 7666

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 6658	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
2	1	eleq2d 2687	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 ∅)))
3		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
4	3	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 𝑦)))
5		oveq2 6658	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
6	5	eleq2d 2687	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))
7		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵))
8	7	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ ∅ ∈ (𝐴 ↑𝑜 𝐵)))
9		0lt1o 7584	. . . . . . 7 ⊢ ∅ ∈ 1𝑜
10		oe0 7602	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
11	9, 10	syl5eleqr 2708	. . . . . 6 ⊢ (𝐴 ∈ On → ∅ ∈ (𝐴 ↑𝑜 ∅))
12	11	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 ∅))
13		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
14		oecl 7617	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
15	13, 14	jca 554	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On))
16		omordi 7646	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
17		om0 7597	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → ((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) = ∅)
18	17	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → (((𝐴 ↑𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
19	18	ad2antlr 763	. . . . . . . . . . . 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 229	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
21	15, 20	sylan 488	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
22		oesuc 7607	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) = ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴))
23	22	eleq2d 2687	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴 ↑𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
24	23	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ (𝐴 ↑𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑𝑜 𝑦) ·𝑜 𝐴)))
25	21, 24	sylibrd 249	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))
26	25	exp31 630	. . . . . . . 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 91	. . . . . 6 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (∅ ∈ (𝐴 ↑𝑜 𝑦) → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦)))))
29	28	impd 447	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴 ↑𝑜 𝑦) → ∅ ∈ (𝐴 ↑𝑜 suc 𝑦))))
30		0ellim 5787	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
31		eqimss2 3658	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑𝑜 ∅) = 1𝑜 → 1𝑜 ⊆ (𝐴 ↑𝑜 ∅))
32	10, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1𝑜 ⊆ (𝐴 ↑𝑜 ∅))
33		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑𝑜 𝑦) = (𝐴 ↑𝑜 ∅))
34	33	sseq2d 3633	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1𝑜 ⊆ (𝐴 ↑𝑜 𝑦) ↔ 1𝑜 ⊆ (𝐴 ↑𝑜 ∅)))
35	34	rspcev 3309	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑥 ∧ 1𝑜 ⊆ (𝐴 ↑𝑜 ∅)) → ∃𝑦 ∈ 𝑥 1𝑜 ⊆ (𝐴 ↑𝑜 𝑦))
36	30, 32, 35	syl2an 494	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝑥 1𝑜 ⊆ (𝐴 ↑𝑜 𝑦))
37		ssiun 4562	. . . . . . . . . . 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 753	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
40		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
41		oelim 7614	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
42	40, 41	mpanlr1 722	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
43	42	anasss 679	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
44	43	an12s 843	. . . . . . . . 9 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
45	39, 44	sseqtr4d 3642	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 ⊆ (𝐴 ↑𝑜 𝑥))
46		limelon 5788	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
47	40, 46	mpan 706	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
48		oecl 7617	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑𝑜 𝑥) ∈ On)
49	48	ancoms 469	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 𝑥) ∈ On)
50	47, 49	sylan 488	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ↑𝑜 𝑥) ∈ On)
51		eloni 5733	. . . . . . . . . 10 ⊢ ((𝐴 ↑𝑜 𝑥) ∈ On → Ord (𝐴 ↑𝑜 𝑥))
52		ordgt0ge1 7577	. . . . . . . . . 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 753	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴 ↑𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴 ↑𝑜 𝑥)))
55	45, 54	mpbird 247	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑𝑜 𝑥))
56	55	ex 450	. . . . . 6 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝑥)))
57	56	a1dd 50	. . . . 5 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 ∅ ∈ (𝐴 ↑𝑜 𝑦) → ∅ ∈ (𝐴 ↑𝑜 𝑥))))
58	2, 4, 6, 8, 12, 29, 57	tfinds3 7064	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵)))
59	58	expd 452	. . 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 448	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑𝑜 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553   ·_𝑜 comu 7558   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oeordi  7667  oeordsuc  7674  oeoelem  7678  oelimcl  7680  oeeui  7682  cantnflt  8569  cnfcom  8597  infxpenc  8841  infxpenc2  8845