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Theorem oen0 7950
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 6930 . . . . . 6 (𝑥 = ∅ → (𝐴o 𝑥) = (𝐴o ∅))
21eleq2d 2845 . . . . 5 (𝑥 = ∅ → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o ∅)))
3 oveq2 6930 . . . . . 6 (𝑥 = 𝑦 → (𝐴o 𝑥) = (𝐴o 𝑦))
43eleq2d 2845 . . . . 5 (𝑥 = 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝑦)))
5 oveq2 6930 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴o 𝑥) = (𝐴o suc 𝑦))
65eleq2d 2845 . . . . 5 (𝑥 = suc 𝑦 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o suc 𝑦)))
7 oveq2 6930 . . . . . 6 (𝑥 = 𝐵 → (𝐴o 𝑥) = (𝐴o 𝐵))
87eleq2d 2845 . . . . 5 (𝑥 = 𝐵 → (∅ ∈ (𝐴o 𝑥) ↔ ∅ ∈ (𝐴o 𝐵)))
9 0lt1o 7868 . . . . . . 7 ∅ ∈ 1o
10 oe0 7886 . . . . . . 7 (𝐴 ∈ On → (𝐴o ∅) = 1o)
119, 10syl5eleqr 2866 . . . . . 6 (𝐴 ∈ On → ∅ ∈ (𝐴o ∅))
1211adantr 474 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o ∅))
13 oecl 7901 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o 𝑦) ∈ On)
14 omordi 7930 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴)))
15 om0 7881 . . . . . . . . . . . . . 14 ((𝐴o 𝑦) ∈ On → ((𝐴o 𝑦) ·o ∅) = ∅)
1615eleq1d 2844 . . . . . . . . . . . . 13 ((𝐴o 𝑦) ∈ On → (((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1716ad2antlr 717 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (((𝐴o 𝑦) ·o ∅) ∈ ((𝐴o 𝑦) ·o 𝐴) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1814, 17sylibd 231 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝐴o 𝑦) ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
1913, 18syldanl 595 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
20 oesuc 7891 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴o suc 𝑦) = ((𝐴o 𝑦) ·o 𝐴))
2120eleq2d 2845 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴o suc 𝑦) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
2221adantr 474 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ (𝐴o suc 𝑦) ↔ ∅ ∈ ((𝐴o 𝑦) ·o 𝐴)))
2319, 22sylibrd 251 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴o 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o suc 𝑦)))
2423exp31 412 . . . . . . . 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 400 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴o 𝑦) → ∅ ∈ (𝐴o suc 𝑦))))
28 0ellim 6038 . . . . . . . . . . . 12 (Lim 𝑥 → ∅ ∈ 𝑥)
29 eqimss2 3877 . . . . . . . . . . . . 13 ((𝐴o ∅) = 1o → 1o ⊆ (𝐴o ∅))
3010, 29syl 17 . . . . . . . . . . . 12 (𝐴 ∈ On → 1o ⊆ (𝐴o ∅))
31 oveq2 6930 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐴o 𝑦) = (𝐴o ∅))
3231sseq2d 3852 . . . . . . . . . . . . 13 (𝑦 = ∅ → (1o ⊆ (𝐴o 𝑦) ↔ 1o ⊆ (𝐴o ∅)))
3332rspcev 3511 . . . . . . . . . . . 12 ((∅ ∈ 𝑥 ∧ 1o ⊆ (𝐴o ∅)) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
3428, 30, 33syl2an 589 . . . . . . . . . . 11 ((Lim 𝑥𝐴 ∈ On) → ∃𝑦𝑥 1o ⊆ (𝐴o 𝑦))
35 ssiun 4795 . . . . . . . . . . 11 (∃𝑦𝑥 1o ⊆ (𝐴o 𝑦) → 1o 𝑦𝑥 (𝐴o 𝑦))
3634, 35syl 17 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → 1o 𝑦𝑥 (𝐴o 𝑦))
3736adantrr 707 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o 𝑦𝑥 (𝐴o 𝑦))
38 vex 3401 . . . . . . . . . . . 12 𝑥 ∈ V
39 oelim 7898 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4038, 39mpanlr1 696 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4140anasss 460 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4241an12s 639 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴o 𝑥) = 𝑦𝑥 (𝐴o 𝑦))
4337, 42sseqtr4d 3861 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1o ⊆ (𝐴o 𝑥))
44 limelon 6039 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
4538, 44mpan 680 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
46 oecl 7901 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴o 𝑥) ∈ On)
4746ancoms 452 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴o 𝑥) ∈ On)
4845, 47sylan 575 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → (𝐴o 𝑥) ∈ On)
49 eloni 5986 . . . . . . . . . 10 ((𝐴o 𝑥) ∈ On → Ord (𝐴o 𝑥))
50 ordgt0ge1 7861 . . . . . . . . . 10 (Ord (𝐴o 𝑥) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5148, 49, 503syl 18 . . . . . . . . 9 ((Lim 𝑥𝐴 ∈ On) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5251adantrr 707 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴o 𝑥) ↔ 1o ⊆ (𝐴o 𝑥)))
5343, 52mpbird 249 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴o 𝑥))
5453ex 403 . . . . . 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 7342 . . . 4 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵)))
5756expd 406 . . 3 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o 𝐵))))
5857com12 32 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴o 𝐵))))
5958imp31 410 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  Vcvv 3398  wss 3792  c0 4141   ciun 4753  Ord word 5975  Oncon0 5976  Lim wlim 5977  suc csuc 5978  (class class class)co 6922  1oc1o 7836   ·o comu 7841  o coe 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-omul 7848  df-oexp 7849
This theorem is referenced by:  oeordi  7951  oeordsuc  7958  oeoelem  7962  oelimcl  7964  oeeui  7966  cantnflt  8866  cnfcom  8894  infxpenc  9174  infxpenc2  9178
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