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Theorem noreson 27720
Description: The restriction of a surreal to an ordinal is still a surreal. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
noreson ((𝐴 No 𝐵 ∈ On) → (𝐴𝐵) ∈ No )

Proof of Theorem noreson
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elno 27705 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2 onin 6417 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵) ∈ On)
3 fresin 6778 . . . . . . . 8 (𝐴:𝑥⟶{1o, 2o} → (𝐴𝐵):(𝑥𝐵)⟶{1o, 2o})
4 feq2 6718 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → ((𝐴𝐵):𝑦⟶{1o, 2o} ↔ (𝐴𝐵):(𝑥𝐵)⟶{1o, 2o}))
54rspcev 3622 . . . . . . . 8 (((𝑥𝐵) ∈ On ∧ (𝐴𝐵):(𝑥𝐵)⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
62, 3, 5syl2an 596 . . . . . . 7 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
76an32s 652 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
87ex 412 . . . . 5 ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o}))
98rexlimiva 3145 . . . 4 (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o}))
109imp 406 . . 3 ((∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
111, 10sylanb 581 . 2 ((𝐴 No 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
12 elno 27705 . 2 ((𝐴𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
1311, 12sylibr 234 1 ((𝐴 No 𝐵 ∈ On) → (𝐴𝐵) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wrex 3068  cin 3962  {cpr 4633  cres 5691  Oncon0 6386  wf 6559  1oc1o 8498  2oc2o 8499   No csur 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ord 6389  df-on 6390  df-fun 6565  df-fn 6566  df-f 6567  df-no 27702
This theorem is referenced by:  sltres  27722  nodenselem6  27749  noresle  27757  nosupbnd1lem1  27768  nosupbnd1lem2  27769  nosupbnd1lem6  27773  nosupbnd1  27774  nosupbnd2lem1  27775  nosupbnd2  27776  noinfbnd1lem1  27783  noinfbnd1lem2  27784  noinfbnd1lem6  27788  noinfbnd1  27789  noinfbnd2lem1  27790  noinfbnd2  27791  nosupinfsep  27792  noetasuplem4  27796  noetainflem4  27800
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