Theorem oen0 8214

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 7166	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
2	1	eleq2d 2900	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o ∅)))
3		oveq2 7166	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
4	3	eleq2d 2900	. . . . 5 ⊢ (𝑥 = 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝑦)))
5		oveq2 7166	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
6	5	eleq2d 2900	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o suc 𝑦)))
7		oveq2 7166	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
8	7	eleq2d 2900	. . . . 5 ⊢ (𝑥 = 𝐵 → (∅ ∈ (𝐴 ↑o 𝑥) ↔ ∅ ∈ (𝐴 ↑o 𝐵)))
9		0lt1o 8131	. . . . . . 7 ⊢ ∅ ∈ 1o
10		oe0 8149	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
11	9, 10	eleqtrrid 2922	. . . . . 6 ⊢ (𝐴 ∈ On → ∅ ∈ (𝐴 ↑o ∅))
12	11	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o ∅))
13		oecl 8164	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
14		omordi 8194	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴 ↑o 𝑦) ·o ∅) ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
15		om0 8144	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↑o 𝑦) ∈ On → ((𝐴 ↑o 𝑦) ·o ∅) = ∅)
16	15	eleq1d 2899	. . . . . . . . . . . . 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 241	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ (𝐴 ↑o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
19	13, 18	syldanl 603	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
20		oesuc 8154	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) = ((𝐴 ↑o 𝑦) ·o 𝐴))
21	20	eleq2d 2900	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
22	21	adantr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ (𝐴 ↑o suc 𝑦) ↔ ∅ ∈ ((𝐴 ↑o 𝑦) ·o 𝐴)))
23	19, 22	sylibrd 261	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴 ↑o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 ↑o suc 𝑦)))
24	23	exp31 422	. . . . . . . 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 413	. . . . 5 ⊢ (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴 ↑o 𝑦) → ∅ ∈ (𝐴 ↑o suc 𝑦))))
28		0ellim 6255	. . . . . . . . . . . 12 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		eqimss2 4026	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↑o ∅) = 1o → 1o ⊆ (𝐴 ↑o ∅))
30	10, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 1o ⊆ (𝐴 ↑o ∅))
31		oveq2 7166	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (𝐴 ↑o 𝑦) = (𝐴 ↑o ∅))
32	31	sseq2d 4001	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (1o ⊆ (𝐴 ↑o 𝑦) ↔ 1o ⊆ (𝐴 ↑o ∅)))
33	32	rspcev 3625	. . . . . . . . . . . 12 ⊢ ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴 ↑o ∅)) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦))
34	28, 30, 33	syl2an 597	. . . . . . . . . . 11 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → ∃𝑦 ∈ 𝑥 1o ⊆ (𝐴 ↑o 𝑦))
35		ssiun 4972	. . . . . . . . . . 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 3499	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
39		oelim 8161	. . . . . . . . . . . 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 469	. . . . . . . . . 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 4010	. . . . . . . 8 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴 ↑o 𝑥))
44		limelon 6256	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
45	38, 44	mpan 688	. . . . . . . . . . 11 ⊢ (Lim 𝑥 → 𝑥 ∈ On)
46		oecl 8164	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
47	46	ancoms 461	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
48	45, 47	sylan 582	. . . . . . . . . 10 ⊢ ((Lim 𝑥 ∧ 𝐴 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
49		eloni 6203	. . . . . . . . . 10 ⊢ ((𝐴 ↑o 𝑥) ∈ On → Ord (𝐴 ↑o 𝑥))
50		ordgt0ge1 8124	. . . . . . . . . 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 259	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴 ↑o 𝑥))
54	53	ex 415	. . . . . 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 7581	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵)))
57	56	expd 418	. . 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 420	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  ∪ ciun 4921  Ord word 6192  Oncon0 6193  Lim wlim 6194  suc csuc 6195  (class class class)co 7158  1_oc1o 8097   ·_o comu 8102   ↑_o coe 8103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-omul 8109  df-oexp 8110

This theorem is referenced by:  oeordi  8215  oeordsuc  8222  oeoelem  8226  oelimcl  8228  oeeui  8230  cantnflt  9137  cnfcom  9165  infxpenc  9446  infxpenc2  9450