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Theorem noextend 27726
Description: Extending a surreal by one sign value results in a new surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
Hypothesis
Ref Expression
noextend.1 𝑋 ∈ {1o, 2o}
Assertion
Ref Expression
noextend (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )

Proof of Theorem noextend
StepHypRef Expression
1 nofun 27709 . . 3 (𝐴 No → Fun 𝐴)
2 dmexg 7924 . . . 4 (𝐴 No → dom 𝐴 ∈ V)
3 noextend.1 . . . 4 𝑋 ∈ {1o, 2o}
4 funsng 6619 . . . 4 ((dom 𝐴 ∈ V ∧ 𝑋 ∈ {1o, 2o}) → Fun {⟨dom 𝐴, 𝑋⟩})
52, 3, 4sylancl 586 . . 3 (𝐴 No → Fun {⟨dom 𝐴, 𝑋⟩})
63elexi 3501 . . . . . 6 𝑋 ∈ V
76dmsnop 6238 . . . . 5 dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
87ineq2i 4225 . . . 4 (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9 nodmord 27713 . . . . . 6 (𝐴 No → Ord dom 𝐴)
10 ordirr 6404 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
119, 10syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
12 disjsn 4716 . . . . 5 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1311, 12sylibr 234 . . . 4 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
148, 13eqtrid 2787 . . 3 (𝐴 No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15 funun 6614 . . 3 (((Fun 𝐴 ∧ Fun {⟨dom 𝐴, 𝑋⟩}) ∧ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅) → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
161, 5, 14, 15syl21anc 838 . 2 (𝐴 No → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
177uneq2i 4175 . . . 4 (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18 dmun 5924 . . . 4 dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19 df-suc 6392 . . . 4 suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
2017, 18, 193eqtr4i 2773 . . 3 dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21 nodmon 27710 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
22 onsuc 7831 . . . 4 (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
2321, 22syl 17 . . 3 (𝐴 No → suc dom 𝐴 ∈ On)
2420, 23eqeltrid 2843 . 2 (𝐴 No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25 rnun 6168 . . . 4 ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26 rnsnopg 6243 . . . . . 6 (dom 𝐴 ∈ V → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
272, 26syl 17 . . . . 5 (𝐴 No → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
2827uneq2d 4178 . . . 4 (𝐴 No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
2925, 28eqtrid 2787 . . 3 (𝐴 No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30 norn 27711 . . . 4 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
31 snssi 4813 . . . . 5 (𝑋 ∈ {1o, 2o} → {𝑋} ⊆ {1o, 2o})
323, 31mp1i 13 . . . 4 (𝐴 No → {𝑋} ⊆ {1o, 2o})
3330, 32unssd 4202 . . 3 (𝐴 No → (ran 𝐴 ∪ {𝑋}) ⊆ {1o, 2o})
3429, 33eqsstrd 4034 . 2 (𝐴 No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o})
35 elno2 27714 . 2 ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No ↔ (Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∧ dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On ∧ ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1o, 2o}))
3616, 24, 34, 35syl3anbrc 1342 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  {cpr 4633  cop 4637  dom cdm 5689  ran crn 5690  Ord word 6385  Oncon0 6386  suc csuc 6388  Fun wfun 6557  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-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-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  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-pss 3983  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-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-no 27702
This theorem is referenced by:  noextendlt  27729  noextendgt  27730  nosupno  27763  nosupbnd1  27774  nosupbnd2lem1  27775  noinfno  27778  noinfbnd1  27789  noinfbnd2lem1  27790
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