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Theorem oen0 8624
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) → ∅ ∈ (𝐴o 𝐵))

Proof of Theorem oen0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . . 6 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
21eleq2d 2827 . . . . 5 (𝑥 = ∅ → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o ∅)))
3 oveq2 7439 . . . . . 6 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
43eleq2d 2827 . . . . 5 (𝑥 = 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝑦)))
5 oveq2 7439 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
65eleq2d 2827 . . . . 5 (𝑥 = suc 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o suc 𝑦)))
7 oveq2 7439 . . . . . 6 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
87eleq2d 2827 . . . . 5 (𝑥 = 𝐵 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝐵)))
9 0lt1o 8542 . . . . . . 7 ∅ ∈ 1o
10 oe0 8560 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
119, 10eleqtrrid 2848 . . . . . 6 (𝐴 ∈ On → ∅ ∈ (𝐴o ∅))
1211adantr 480 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o ∅))
13 oecl 8575 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
14 omordi 8604 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴)))
15 om0 8555 . . . . . . . . . . . . . 14 ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o ∅) = ∅)
1615eleq1d 2826 . . . . . . . . . . . . 13 ((𝐴o 𝑦) ∈ On → (((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1716ad2antlr 727 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1814, 17sylibd 239 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1913, 18syldanl 602 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
20 oesuc 8565 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2120eleq2d 2827 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴o suc 𝑦) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
2221adantr 480 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ (𝐴o suc 𝑦) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
2319, 22sylibrd 259 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o suc 𝑦)))
2423exp31 419 . . . . . . . 8 (𝐴 ∈ On → (𝑦 ∈ On → (∅ ∈ (𝐴o 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o suc 𝑦)))))
2524com12 32 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ (𝐴o 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o suc 𝑦)))))
2625com34 91 . . . . . 6 (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (∅ ∈ (𝐴o 𝑦) → ∅ ∈ (𝐴o suc 𝑦)))))
2726impd 410 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴o 𝑦) → ∅ ∈ (𝐴o suc 𝑦))))
28 0ellim 6447 . . . . . . . . . . . 12 (Lim 𝑥 → ∅ ∈ 𝑥)
29 eqimss2 4043 . . . . . . . . . . . . 13 ((𝐴o ∅) = 1o → 1o ⊆ (𝐴o ∅))
3010, 29syl 17 . . . . . . . . . . . 12 (𝐴 ∈ On → 1o ⊆ (𝐴o ∅))
31 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐴o 𝑦) = (𝐴o ∅))
3231sseq2d 4016 . . . . . . . . . . . . 13 (𝑦 = ∅ → (1o ⊆ (𝐴o 𝑦) ↔ 1o ⊆ (𝐴o ∅)))
3332rspcev 3622 . . . . . . . . . . . 12 ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴o ∅)) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
3428, 30, 33syl2an 596 . . . . . . . . . . 11 ((Lim 𝑥𝐴 ∈ On) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
35 ssiun 5046 . . . . . . . . . . 11 (∃𝑦𝑥 1o ⊆ (𝐴o 𝑦) → 1o 𝑦𝑥 (𝐴o 𝑦))
3634, 35syl 17 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → 1o 𝑦𝑥 (𝐴o 𝑦))
3736adantrr 717 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o 𝑦𝑥 (𝐴o 𝑦))
38 vex 3484 . . . . . . . . . . . 12 𝑥 ∈ V
39 oelim 8572 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4038, 39mpanlr1 706 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4140anasss 466 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4241an12s 649 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4337, 42sseqtrrd 4021 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴o 𝑥))
44 limelon 6448 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
4538, 44mpan 690 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
46 oecl 8575 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴o 𝑥) ∈ On)
4746ancoms 458 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴o 𝑥) ∈ On)
4845, 47sylan 580 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → (𝐴o 𝑥) ∈ On)
49 eloni 6394 . . . . . . . . . 10 ((𝐴o 𝑥) ∈ On → Ord (𝐴o 𝑥))
50 ordgt0ge1 8531 . . . . . . . . . 10 (Ord (𝐴o 𝑥) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5148, 49, 503syl 18 . . . . . . . . 9 ((Lim 𝑥𝐴 ∈ On) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5251adantrr 717 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5343, 52mpbird 257 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴o 𝑥))
5453ex 412 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝑥)))
5554a1dd 50 . . . . 5 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 ∅ ∈ (𝐴o 𝑦) → ∅ ∈ (𝐴o 𝑥))))
562, 4, 6, 8, 12, 27, 55tfinds3 7886 . . . 4 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵)))
5756expd 415 . . 3 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o 𝐵))))
5857com12 32 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o 𝐵))))
5958imp31 417 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333   ciun 4991  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431  1oc1o 8499   ·o comu 8504  o coe 8505
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by:  oeordi  8625  oeordsuc  8632  oeoelem  8636  oelimcl  8638  oeeui  8640  cantnflt  9712  cnfcom  9740  infxpenc  10058  infxpenc2  10062  onexoegt  43256  cantnftermord  43333  oacl2g  43343  onmcl  43344  omabs2  43345  omcl2  43346  ofoaf  43368  ofoafo  43369
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