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Theorem noreson 31514
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 31500 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
2 onin 5713 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵) ∈ On)
3 fresin 6030 . . . . . . . 8 (𝐴:𝑥⟶{1𝑜, 2𝑜} → (𝐴𝐵):(𝑥𝐵)⟶{1𝑜, 2𝑜})
4 feq2 5984 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → ((𝐴𝐵):𝑦⟶{1𝑜, 2𝑜} ↔ (𝐴𝐵):(𝑥𝐵)⟶{1𝑜, 2𝑜}))
54rspcev 3295 . . . . . . . 8 (((𝑥𝐵) ∈ On ∧ (𝐴𝐵):(𝑥𝐵)⟶{1𝑜, 2𝑜}) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜})
62, 3, 5syl2an 494 . . . . . . 7 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1𝑜, 2𝑜}) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜})
76an32s 845 . . . . . 6 (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1𝑜, 2𝑜}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜})
87ex 450 . . . . 5 ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1𝑜, 2𝑜}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜}))
98rexlimiva 3021 . . . 4 (∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜}))
109imp 445 . . 3 ((∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜})
111, 10sylanb 489 . 2 ((𝐴 No 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜})
12 elno 31500 . 2 ((𝐴𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴𝐵):𝑦⟶{1𝑜, 2𝑜})
1311, 12sylibr 224 1 ((𝐴 No 𝐵 ∈ On) → (𝐴𝐵) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wrex 2908  cin 3554  {cpr 4150  cres 5076  Oncon0 5682  wf 5843  1𝑜c1o 7498  2𝑜c2o 7499   No csur 31494
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-no 31497
This theorem is referenced by:  sltres  31518  noreslege  31571
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