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Theorem noreson 33277
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 33263 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2 onin 6190 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵) ∈ On)
3 fresin 6521 . . . . . . . 8 (𝐴:𝑥⟶{1o, 2o} → (𝐴𝐵):(𝑥𝐵)⟶{1o, 2o})
4 feq2 6469 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → ((𝐴𝐵):𝑦⟶{1o, 2o} ↔ (𝐴𝐵):(𝑥𝐵)⟶{1o, 2o}))
54rspcev 3571 . . . . . . . 8 (((𝑥𝐵) ∈ On ∧ (𝐴𝐵):(𝑥𝐵)⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
62, 3, 5syl2an 598 . . . . . . 7 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
76an32s 651 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
87ex 416 . . . . 5 ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o}))
98rexlimiva 3240 . . . 4 (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o}))
109imp 410 . . 3 ((∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
111, 10sylanb 584 . 2 ((𝐴 No 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
12 elno 33263 . 2 ((𝐴𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1o, 2o})
1311, 12sylibr 237 1 ((𝐴 No 𝐵 ∈ On) → (𝐴𝐵) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  wrex 3107  cin 3880  {cpr 4527  cres 5521  Oncon0 6159  wf 6320  1oc1o 8078  2oc2o 8079   No csur 33257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-no 33260
This theorem is referenced by:  sltres  33279  nodenselem6  33303  noresle  33310  nosupbnd1lem1  33318  nosupbnd1lem2  33319  nosupbnd1lem6  33323  nosupbnd1  33324  nosupbnd2lem1  33325  nosupbnd2  33326  noetalem3  33329
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