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Theorem oen0 7711
 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) → ∅ ∈ (𝐴𝑜 𝐵))

Proof of Theorem oen0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . . . 6 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
21eleq2d 2716 . . . . 5 (𝑥 = ∅ → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 ∅)))
3 oveq2 6698 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
43eleq2d 2716 . . . . 5 (𝑥 = 𝑦 → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 𝑦)))
5 oveq2 6698 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
65eleq2d 2716 . . . . 5 (𝑥 = suc 𝑦 → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 suc 𝑦)))
7 oveq2 6698 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
87eleq2d 2716 . . . . 5 (𝑥 = 𝐵 → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 𝐵)))
9 0lt1o 7629 . . . . . . 7 ∅ ∈ 1𝑜
10 oe0 7647 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
119, 10syl5eleqr 2737 . . . . . 6 (𝐴 ∈ On → ∅ ∈ (𝐴𝑜 ∅))
1211adantr 480 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 ∅))
13 simpl 472 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
14 oecl 7662 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1513, 14jca 553 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ∈ On ∧ (𝐴𝑜 𝑦) ∈ On))
16 omordi 7691 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ 𝐴 → ((𝐴𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
17 om0 7642 . . . . . . . . . . . . . 14 ((𝐴𝑜 𝑦) ∈ On → ((𝐴𝑜 𝑦) ·𝑜 ∅) = ∅)
1817eleq1d 2715 . . . . . . . . . . . . 13 ((𝐴𝑜 𝑦) ∈ On → (((𝐴𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ↔ ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
1918ad2antlr 763 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (((𝐴𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ↔ ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
2016, 19sylibd 229 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝐴𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
2115, 20sylan 487 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
22 oesuc 7652 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq2d 2716 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
2423adantr 480 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ (𝐴𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
2521, 24sylibrd 249 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 suc 𝑦)))
2625exp31 629 . . . . . . . 8 (𝐴 ∈ On → (𝑦 ∈ On → (∅ ∈ (𝐴𝑜 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 suc 𝑦)))))
2726com12 32 . . . . . . 7 (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ (𝐴𝑜 𝑦) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 suc 𝑦)))))
2827com34 91 . . . . . 6 (𝑦 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → (∅ ∈ (𝐴𝑜 𝑦) → ∅ ∈ (𝐴𝑜 suc 𝑦)))))
2928impd 446 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴𝑜 𝑦) → ∅ ∈ (𝐴𝑜 suc 𝑦))))
30 0ellim 5825 . . . . . . . . . . . 12 (Lim 𝑥 → ∅ ∈ 𝑥)
31 eqimss2 3691 . . . . . . . . . . . . 13 ((𝐴𝑜 ∅) = 1𝑜 → 1𝑜 ⊆ (𝐴𝑜 ∅))
3210, 31syl 17 . . . . . . . . . . . 12 (𝐴 ∈ On → 1𝑜 ⊆ (𝐴𝑜 ∅))
33 oveq2 6698 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐴𝑜 𝑦) = (𝐴𝑜 ∅))
3433sseq2d 3666 . . . . . . . . . . . . 13 (𝑦 = ∅ → (1𝑜 ⊆ (𝐴𝑜 𝑦) ↔ 1𝑜 ⊆ (𝐴𝑜 ∅)))
3534rspcev 3340 . . . . . . . . . . . 12 ((∅ ∈ 𝑥 ∧ 1𝑜 ⊆ (𝐴𝑜 ∅)) → ∃𝑦𝑥 1𝑜 ⊆ (𝐴𝑜 𝑦))
3630, 32, 35syl2an 493 . . . . . . . . . . 11 ((Lim 𝑥𝐴 ∈ On) → ∃𝑦𝑥 1𝑜 ⊆ (𝐴𝑜 𝑦))
37 ssiun 4594 . . . . . . . . . . 11 (∃𝑦𝑥 1𝑜 ⊆ (𝐴𝑜 𝑦) → 1𝑜 𝑦𝑥 (𝐴𝑜 𝑦))
3836, 37syl 17 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → 1𝑜 𝑦𝑥 (𝐴𝑜 𝑦))
3938adantrr 753 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 𝑦𝑥 (𝐴𝑜 𝑦))
40 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
41 oelim 7659 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4240, 41mpanlr1 722 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4342anasss 680 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4443an12s 860 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4539, 44sseqtr4d 3675 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 ⊆ (𝐴𝑜 𝑥))
46 limelon 5826 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
4740, 46mpan 706 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
48 oecl 7662 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
4948ancoms 468 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
5047, 49sylan 487 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
51 eloni 5771 . . . . . . . . . 10 ((𝐴𝑜 𝑥) ∈ On → Ord (𝐴𝑜 𝑥))
52 ordgt0ge1 7622 . . . . . . . . . 10 (Ord (𝐴𝑜 𝑥) → (∅ ∈ (𝐴𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴𝑜 𝑥)))
5350, 51, 523syl 18 . . . . . . . . 9 ((Lim 𝑥𝐴 ∈ On) → (∅ ∈ (𝐴𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴𝑜 𝑥)))
5453adantrr 753 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴𝑜 𝑥)))
5545, 54mpbird 247 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴𝑜 𝑥))
5655ex 449 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝑥)))
5756a1dd 50 . . . . 5 (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦𝑥 ∅ ∈ (𝐴𝑜 𝑦) → ∅ ∈ (𝐴𝑜 𝑥))))
582, 4, 6, 8, 12, 29, 57tfinds3 7106 . . . 4 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵)))
5958expd 451 . . 3 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 𝐵))))
6059com12 32 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 𝐵))))
6160imp31 447 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ∪ ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   ·𝑜 comu 7603   ↑𝑜 coe 7604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-oexp 7611 This theorem is referenced by:  oeordi  7712  oeordsuc  7719  oeoelem  7723  oelimcl  7725  oeeui  7727  cantnflt  8607  cnfcom  8635  infxpenc  8879  infxpenc2  8883
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