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Theorem oen0 8600
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 7422 . . . . . 6 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
21eleq2d 2814 . . . . 5 (𝑥 = ∅ → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o ∅)))
3 oveq2 7422 . . . . . 6 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
43eleq2d 2814 . . . . 5 (𝑥 = 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝑦)))
5 oveq2 7422 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
65eleq2d 2814 . . . . 5 (𝑥 = suc 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o suc 𝑦)))
7 oveq2 7422 . . . . . 6 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
87eleq2d 2814 . . . . 5 (𝑥 = 𝐵 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝐵)))
9 0lt1o 8518 . . . . . . 7 ∅ ∈ 1o
10 oe0 8536 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
119, 10eleqtrrid 2835 . . . . . 6 (𝐴 ∈ On → ∅ ∈ (𝐴o ∅))
1211adantr 480 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o ∅))
13 oecl 8551 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
14 omordi 8580 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴)))
15 om0 8531 . . . . . . . . . . . . . 14 ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o ∅) = ∅)
1615eleq1d 2813 . . . . . . . . . . . . 13 ((𝐴o 𝑦) ∈ On → (((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1716ad2antlr 726 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1814, 17sylibd 238 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1913, 18syldanl 601 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
20 oesuc 8541 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2120eleq2d 2814 . . . . . . . . . . 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 6426 . . . . . . . . . . . 12 (Lim 𝑥 → ∅ ∈ 𝑥)
29 eqimss2 4037 . . . . . . . . . . . . 13 ((𝐴o ∅) = 1o → 1o ⊆ (𝐴o ∅))
3010, 29syl 17 . . . . . . . . . . . 12 (𝐴 ∈ On → 1o ⊆ (𝐴o ∅))
31 oveq2 7422 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐴o 𝑦) = (𝐴o ∅))
3231sseq2d 4010 . . . . . . . . . . . . 13 (𝑦 = ∅ → (1o ⊆ (𝐴o 𝑦) ↔ 1o ⊆ (𝐴o ∅)))
3332rspcev 3607 . . . . . . . . . . . 12 ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴o ∅)) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
3428, 30, 33syl2an 595 . . . . . . . . . . 11 ((Lim 𝑥𝐴 ∈ On) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
35 ssiun 5043 . . . . . . . . . . 11 (∃𝑦𝑥 1o ⊆ (𝐴o 𝑦) → 1o 𝑦𝑥 (𝐴o 𝑦))
3634, 35syl 17 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → 1o 𝑦𝑥 (𝐴o 𝑦))
3736adantrr 716 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o 𝑦𝑥 (𝐴o 𝑦))
38 vex 3473 . . . . . . . . . . . 12 𝑥 ∈ V
39 oelim 8548 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4038, 39mpanlr1 705 . . . . . . . . . . 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 648 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4337, 42sseqtrrd 4019 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴o 𝑥))
44 limelon 6427 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
4538, 44mpan 689 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
46 oecl 8551 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴o 𝑥) ∈ On)
4746ancoms 458 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴o 𝑥) ∈ On)
4845, 47sylan 579 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → (𝐴o 𝑥) ∈ On)
49 eloni 6373 . . . . . . . . . 10 ((𝐴o 𝑥) ∈ On → Ord (𝐴o 𝑥))
50 ordgt0ge1 8507 . . . . . . . . . 10 (Ord (𝐴o 𝑥) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5148, 49, 503syl 18 . . . . . . . . 9 ((Lim 𝑥𝐴 ∈ On) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5251adantrr 716 . . . . . . . 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 7863 . . . 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 205  wa 395   = wceq 1534  wcel 2099  wral 3056  wrex 3065  Vcvv 3469  wss 3944  c0 4318   ciun 4991  Ord word 6362  Oncon0 6363  Lim wlim 6364  suc csuc 6365  (class class class)co 7414  1oc1o 8473   ·o comu 8478  o coe 8479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-oexp 8486
This theorem is referenced by:  oeordi  8601  oeordsuc  8608  oeoelem  8612  oelimcl  8614  oeeui  8616  cantnflt  9687  cnfcom  9715  infxpenc  10033  infxpenc2  10037  onexoegt  42595  cantnftermord  42672  oacl2g  42682  onmcl  42683  omabs2  42684  omcl2  42685  ofoaf  42707  ofoafo  42708
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