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Theorem oen0 7612
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 6613 . . . . . 6 (𝑥 = ∅ → (𝐴𝑜 𝑥) = (𝐴𝑜 ∅))
21eleq2d 2689 . . . . 5 (𝑥 = ∅ → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 ∅)))
3 oveq2 6613 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝑦))
43eleq2d 2689 . . . . 5 (𝑥 = 𝑦 → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 𝑦)))
5 oveq2 6613 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴𝑜 𝑥) = (𝐴𝑜 suc 𝑦))
65eleq2d 2689 . . . . 5 (𝑥 = suc 𝑦 → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 suc 𝑦)))
7 oveq2 6613 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝑜 𝑥) = (𝐴𝑜 𝐵))
87eleq2d 2689 . . . . 5 (𝑥 = 𝐵 → (∅ ∈ (𝐴𝑜 𝑥) ↔ ∅ ∈ (𝐴𝑜 𝐵)))
9 0lt1o 7530 . . . . . . 7 ∅ ∈ 1𝑜
10 oe0 7548 . . . . . . 7 (𝐴 ∈ On → (𝐴𝑜 ∅) = 1𝑜)
119, 10syl5eleqr 2711 . . . . . 6 (𝐴 ∈ On → ∅ ∈ (𝐴𝑜 ∅))
1211adantr 481 . . . . 5 ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 ∅))
13 simpl 473 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
14 oecl 7563 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 𝑦) ∈ On)
1513, 14jca 554 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ∈ On ∧ (𝐴𝑜 𝑦) ∈ On))
16 omordi 7592 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝐴𝑜 𝑦) ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ 𝐴 → ((𝐴𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
17 om0 7543 . . . . . . . . . . . . . 14 ((𝐴𝑜 𝑦) ∈ On → ((𝐴𝑜 𝑦) ·𝑜 ∅) = ∅)
1817eleq1d 2688 . . . . . . . . . . . . 13 ((𝐴𝑜 𝑦) ∈ On → (((𝐴𝑜 𝑦) ·𝑜 ∅) ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴) ↔ ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
1918ad2antlr 762 . . . . . . . . . . . 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 488 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ ∅ ∈ (𝐴𝑜 𝑦)) → (∅ ∈ 𝐴 → ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
22 oesuc 7553 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝑜 suc 𝑦) = ((𝐴𝑜 𝑦) ·𝑜 𝐴))
2322eleq2d 2689 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (∅ ∈ (𝐴𝑜 suc 𝑦) ↔ ∅ ∈ ((𝐴𝑜 𝑦) ·𝑜 𝐴)))
2423adantr 481 . . . . . . . . . 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 447 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∅ ∈ (𝐴𝑜 𝑦) → ∅ ∈ (𝐴𝑜 suc 𝑦))))
30 0ellim 5749 . . . . . . . . . . . 12 (Lim 𝑥 → ∅ ∈ 𝑥)
31 eqimss2 3642 . . . . . . . . . . . . 13 ((𝐴𝑜 ∅) = 1𝑜 → 1𝑜 ⊆ (𝐴𝑜 ∅))
3210, 31syl 17 . . . . . . . . . . . 12 (𝐴 ∈ On → 1𝑜 ⊆ (𝐴𝑜 ∅))
33 oveq2 6613 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐴𝑜 𝑦) = (𝐴𝑜 ∅))
3433sseq2d 3617 . . . . . . . . . . . . 13 (𝑦 = ∅ → (1𝑜 ⊆ (𝐴𝑜 𝑦) ↔ 1𝑜 ⊆ (𝐴𝑜 ∅)))
3534rspcev 3300 . . . . . . . . . . . 12 ((∅ ∈ 𝑥 ∧ 1𝑜 ⊆ (𝐴𝑜 ∅)) → ∃𝑦𝑥 1𝑜 ⊆ (𝐴𝑜 𝑦))
3630, 32, 35syl2an 494 . . . . . . . . . . 11 ((Lim 𝑥𝐴 ∈ On) → ∃𝑦𝑥 1𝑜 ⊆ (𝐴𝑜 𝑦))
37 ssiun 4533 . . . . . . . . . . 11 (∃𝑦𝑥 1𝑜 ⊆ (𝐴𝑜 𝑦) → 1𝑜 𝑦𝑥 (𝐴𝑜 𝑦))
3836, 37syl 17 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → 1𝑜 𝑦𝑥 (𝐴𝑜 𝑦))
3938adantrr 752 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 𝑦𝑥 (𝐴𝑜 𝑦))
40 vex 3194 . . . . . . . . . . . 12 𝑥 ∈ V
41 oelim 7560 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4240, 41mpanlr1 721 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4342anasss 678 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4443an12s 842 . . . . . . . . 9 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴𝑜 𝑥) = 𝑦𝑥 (𝐴𝑜 𝑦))
4539, 44sseqtr4d 3626 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 1𝑜 ⊆ (𝐴𝑜 𝑥))
46 limelon 5750 . . . . . . . . . . . 12 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
4740, 46mpan 705 . . . . . . . . . . 11 (Lim 𝑥𝑥 ∈ On)
48 oecl 7563 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
4948ancoms 469 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐴 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
5047, 49sylan 488 . . . . . . . . . 10 ((Lim 𝑥𝐴 ∈ On) → (𝐴𝑜 𝑥) ∈ On)
51 eloni 5695 . . . . . . . . . 10 ((𝐴𝑜 𝑥) ∈ On → Ord (𝐴𝑜 𝑥))
52 ordgt0ge1 7523 . . . . . . . . . 10 (Ord (𝐴𝑜 𝑥) → (∅ ∈ (𝐴𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴𝑜 𝑥)))
5350, 51, 523syl 18 . . . . . . . . 9 ((Lim 𝑥𝐴 ∈ On) → (∅ ∈ (𝐴𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴𝑜 𝑥)))
5453adantrr 752 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∅ ∈ (𝐴𝑜 𝑥) ↔ 1𝑜 ⊆ (𝐴𝑜 𝑥)))
5545, 54mpbird 247 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → ∅ ∈ (𝐴𝑜 𝑥))
5655ex 450 . . . . . 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 7012 . . . 4 (𝐵 ∈ On → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵)))
5958expd 452 . . 3 (𝐵 ∈ On → (𝐴 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 𝐵))))
6059com12 32 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (∅ ∈ 𝐴 → ∅ ∈ (𝐴𝑜 𝐵))))
6160imp31 448 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  wss 3560  c0 3896   ciun 4490  Ord word 5684  Oncon0 5685  Lim wlim 5686  suc csuc 5687  (class class class)co 6605  1𝑜c1o 7499   ·𝑜 comu 7504  𝑜 coe 7505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511  df-oexp 7512
This theorem is referenced by:  oeordi  7613  oeordsuc  7620  oeoelem  7624  oelimcl  7626  oeeui  7628  cantnflt  8514  cnfcom  8542  infxpenc  8786  infxpenc2  8790
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