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Theorem oen0 8622
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 7438 . . . . . 6 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
21eleq2d 2824 . . . . 5 (𝑥 = ∅ → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o ∅)))
3 oveq2 7438 . . . . . 6 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
43eleq2d 2824 . . . . 5 (𝑥 = 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝑦)))
5 oveq2 7438 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
65eleq2d 2824 . . . . 5 (𝑥 = suc 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o suc 𝑦)))
7 oveq2 7438 . . . . . 6 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
87eleq2d 2824 . . . . 5 (𝑥 = 𝐵 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝐵)))
9 0lt1o 8540 . . . . . . 7 ∅ ∈ 1o
10 oe0 8558 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
119, 10eleqtrrid 2845 . . . . . 6 (𝐴 ∈ On → ∅ ∈ (𝐴o ∅))
1211adantr 480 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o ∅))
13 oecl 8573 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
14 omordi 8602 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴)))
15 om0 8553 . . . . . . . . . . . . . 14 ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o ∅) = ∅)
1615eleq1d 2823 . . . . . . . . . . . . 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 8563 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2120eleq2d 2824 . . . . . . . . . . 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 6448 . . . . . . . . . . . 12 (Lim 𝑥 → ∅ ∈ 𝑥)
29 eqimss2 4054 . . . . . . . . . . . . 13 ((𝐴o ∅) = 1o → 1o ⊆ (𝐴o ∅))
3010, 29syl 17 . . . . . . . . . . . 12 (𝐴 ∈ On → 1o ⊆ (𝐴o ∅))
31 oveq2 7438 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐴o 𝑦) = (𝐴o ∅))
3231sseq2d 4027 . . . . . . . . . . . . 13 (𝑦 = ∅ → (1o ⊆ (𝐴o 𝑦) ↔ 1o ⊆ (𝐴o ∅)))
3332rspcev 3621 . . . . . . . . . . . 12 ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴o ∅)) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
3428, 30, 33syl2an 596 . . . . . . . . . . 11 ((Lim 𝑥𝐴 ∈ On) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
35 ssiun 5050 . . . . . . . . . . 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 3481 . . . . . . . . . . . 12 𝑥 ∈ V
39 oelim 8570 . . . . . . . . . . . 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 4036 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴o 𝑥))
44 limelon 6449 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
4538, 44mpan 690 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
46 oecl 8573 . . . . . . . . . . . 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 6395 . . . . . . . . . 10 ((𝐴o 𝑥) ∈ On → Ord (𝐴o 𝑥))
50 ordgt0ge1 8529 . . . . . . . . . 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 7885 . . . 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 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338   ciun 4995  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430  1oc1o 8497   ·o comu 8502  o coe 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-oexp 8510
This theorem is referenced by:  oeordi  8623  oeordsuc  8630  oeoelem  8634  oelimcl  8636  oeeui  8638  cantnflt  9709  cnfcom  9737  infxpenc  10055  infxpenc2  10059  onexoegt  43232  cantnftermord  43309  oacl2g  43319  onmcl  43320  omabs2  43321  omcl2  43322  ofoaf  43344  ofoafo  43345
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