Theorem noreson 27705

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) → (𝐴 ↾ 𝐵) ∈ No )

Proof of Theorem noreson

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elno 27690	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2		onin 6415	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∩ 𝐵) ∈ On)
3		fresin 6777	. . . . . . . 8 ⊢ (𝐴:𝑥⟶{1o, 2o} → (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o})
4		feq2 6717	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ↾ 𝐵):𝑦⟶{1o, 2o} ↔ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}))
5	4	rspcev 3622	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝐵) ∈ On ∧ (𝐴 ↾ 𝐵):(𝑥 ∩ 𝐵)⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
6	2, 3, 5	syl2an 596	. . . . . . 7 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴:𝑥⟶{1o, 2o}) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
7	6	an32s 652	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
8	7	ex 412	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝐴:𝑥⟶{1o, 2o}) → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}))
9	8	rexlimiva 3147	. . . 4 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → (𝐵 ∈ On → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o}))
10	9	imp 406	. . 3 ⊢ ((∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
11	1, 10	sylanb 581	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
12		elno 27690	. 2 ⊢ ((𝐴 ↾ 𝐵) ∈ No ↔ ∃𝑦 ∈ On (𝐴 ↾ 𝐵):𝑦⟶{1o, 2o})
13	11, 12	sylibr 234	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ↾ 𝐵) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∃wrex 3070   ∩ cin 3950  {cpr 4628   ↾ cres 5687  Oncon0 6384  ⟶wf 6557  1_oc1o 8499  2_oc2o 8500   No csur 27684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ord 6387  df-on 6388  df-fun 6563  df-fn 6564  df-f 6565  df-no 27687

This theorem is referenced by:  sltres  27707  nodenselem6  27734  noresle  27742  nosupbnd1lem1  27753  nosupbnd1lem2  27754  nosupbnd1lem6  27758  nosupbnd1  27759  nosupbnd2lem1  27760  nosupbnd2  27761  noinfbnd1lem1  27768  noinfbnd1lem2  27769  noinfbnd1lem6  27773  noinfbnd1  27774  noinfbnd2lem1  27775  noinfbnd2  27776  nosupinfsep  27777  noetasuplem4  27781  noetainflem4  27785