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Theorem noxpsgn 31554
 Description: The Cartesian product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.)
Hypothesis
Ref Expression
noxpsgn.1 𝑋 ∈ {1𝑜, 2𝑜}
Assertion
Ref Expression
noxpsgn (𝐴 ∈ On → (𝐴 × {𝑋}) ∈ No )

Proof of Theorem noxpsgn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 noxpsgn.1 . . . 4 𝑋 ∈ {1𝑜, 2𝑜}
21fconst6 6057 . . 3 (𝐴 × {𝑋}):𝐴⟶{1𝑜, 2𝑜}
3 feq2 5989 . . . 4 (𝑥 = 𝐴 → ((𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜} ↔ (𝐴 × {𝑋}):𝐴⟶{1𝑜, 2𝑜}))
43rspcev 3298 . . 3 ((𝐴 ∈ On ∧ (𝐴 × {𝑋}):𝐴⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On (𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜})
52, 4mpan2 706 . 2 (𝐴 ∈ On → ∃𝑥 ∈ On (𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜})
6 elno 31535 . 2 ((𝐴 × {𝑋}) ∈ No ↔ ∃𝑥 ∈ On (𝐴 × {𝑋}):𝑥⟶{1𝑜, 2𝑜})
75, 6sylibr 224 1 (𝐴 ∈ On → (𝐴 × {𝑋}) ∈ No )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  ∃wrex 2908  {csn 4153  {cpr 4155   × cxp 5077  Oncon0 5687  ⟶wf 5848  1𝑜c1o 7505  2𝑜c2o 7506   No csur 31529 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 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872 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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-no 31532 This theorem is referenced by:  noxp1o  31555  noxp2o  31556  nobndlem3  31592
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