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Theorem noxp1o 27723
Description: The Cartesian product of an ordinal and {1o} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)
Assertion
Ref Expression
noxp1o (𝐴 ∈ On → (𝐴 × {1o}) ∈ No )

Proof of Theorem noxp1o
StepHypRef Expression
1 1oex 8515 . . . . . 6 1o ∈ V
21prid1 4767 . . . . 5 1o ∈ {1o, 2o}
32fconst6 6799 . . . 4 (𝐴 × {1o}):𝐴⟶{1o, 2o}
41snnz 4781 . . . . . 6 {1o} ≠ ∅
5 dmxp 5942 . . . . . 6 ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴)
64, 5ax-mp 5 . . . . 5 dom (𝐴 × {1o}) = 𝐴
76feq2i 6729 . . . 4 ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o})
83, 7mpbir 231 . . 3 (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o}
98a1i 11 . 2 (𝐴 ∈ On → (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o})
106eleq1i 2830 . . 3 (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On)
1110biimpri 228 . 2 (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On)
12 elno3 27715 . 2 ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On))
139, 11, 12sylanbrc 583 1 (𝐴 ∈ On → (𝐴 × {1o}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  c0 4339  {csn 4631  {cpr 4633   × cxp 5687  dom cdm 5689  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-10 2139  ax-11 2155  ax-12 2175  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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  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-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-1o 8505  df-no 27702
This theorem is referenced by:  bdayfo  27737
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