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Theorem noxp1o 27709
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 8517 . . . . . 6 1o ∈ V
21prid1 4761 . . . . 5 1o ∈ {1o, 2o}
32fconst6 6797 . . . 4 (𝐴 × {1o}):𝐴⟶{1o, 2o}
41snnz 4775 . . . . . 6 {1o} ≠ ∅
5 dmxp 5938 . . . . . 6 ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴)
64, 5ax-mp 5 . . . . 5 dom (𝐴 × {1o}) = 𝐴
76feq2i 6727 . . . 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 2831 . . 3 (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On)
1110biimpri 228 . 2 (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On)
12 elno3 27701 . 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 1539  wcel 2107  wne 2939  c0 4332  {csn 4625  {cpr 4627   × cxp 5682  dom cdm 5684  Oncon0 6383  wf 6556  1oc1o 8500  2oc2o 8501   No csur 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-suc 6389  df-fun 6562  df-fn 6563  df-f 6564  df-1o 8507  df-no 27688
This theorem is referenced by:  bdayfo  27723
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