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Theorem noxp1o 27034
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 8426 . . . . . 6 1o ∈ V
21prid1 4727 . . . . 5 1o ∈ {1o, 2o}
32fconst6 6736 . . . 4 (𝐴 × {1o}):𝐴⟶{1o, 2o}
41snnz 4741 . . . . . 6 {1o} ≠ ∅
5 dmxp 5888 . . . . . 6 ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴)
64, 5ax-mp 5 . . . . 5 dom (𝐴 × {1o}) = 𝐴
76feq2i 6664 . . . 4 ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o})
83, 7mpbir 230 . . 3 (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o}
98a1i 11 . 2 (𝐴 ∈ On → (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o})
106eleq1i 2825 . . 3 (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On)
1110biimpri 227 . 2 (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On)
12 elno3 27026 . 2 ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On))
139, 11, 12sylanbrc 584 1 (𝐴 ∈ On → (𝐴 × {1o}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2940  c0 4286  {csn 4590  {cpr 4592   × cxp 5635  dom cdm 5637  Oncon0 6321  wf 6496  1oc1o 8409  2oc2o 8410   No csur 27011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1o 8416  df-no 27014
This theorem is referenced by:  bdayfo  27048
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