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Theorem noextend 31944
 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 31927 . . 3 (𝐴 No → Fun 𝐴)
2 dmexg 7139 . . . 4 (𝐴 No → dom 𝐴 ∈ V)
3 noextend.1 . . . 4 𝑋 ∈ {1𝑜, 2𝑜}
4 funsng 5975 . . . 4 ((dom 𝐴 ∈ V ∧ 𝑋 ∈ {1𝑜, 2𝑜}) → Fun {⟨dom 𝐴, 𝑋⟩})
52, 3, 4sylancl 695 . . 3 (𝐴 No → Fun {⟨dom 𝐴, 𝑋⟩})
63elexi 3244 . . . . . 6 𝑋 ∈ V
76dmsnop 5645 . . . . 5 dom {⟨dom 𝐴, 𝑋⟩} = {dom 𝐴}
87ineq2i 3844 . . . 4 (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∩ {dom 𝐴})
9 nodmord 31931 . . . . . 6 (𝐴 No → Ord dom 𝐴)
10 ordirr 5779 . . . . . 6 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
119, 10syl 17 . . . . 5 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
12 disjsn 4278 . . . . 5 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1311, 12sylibr 224 . . . 4 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
148, 13syl5eq 2697 . . 3 (𝐴 No → (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅)
15 funun 5970 . . 3 (((Fun 𝐴 ∧ Fun {⟨dom 𝐴, 𝑋⟩}) ∧ (dom 𝐴 ∩ dom {⟨dom 𝐴, 𝑋⟩}) = ∅) → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
161, 5, 14, 15syl21anc 1365 . 2 (𝐴 No → Fun (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}))
177uneq2i 3797 . . . 4 (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ {dom 𝐴})
18 dmun 5363 . . . 4 dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (dom 𝐴 ∪ dom {⟨dom 𝐴, 𝑋⟩})
19 df-suc 5767 . . . 4 suc dom 𝐴 = (dom 𝐴 ∪ {dom 𝐴})
2017, 18, 193eqtr4i 2683 . . 3 dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = suc dom 𝐴
21 nodmon 31928 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
22 suceloni 7055 . . . 4 (dom 𝐴 ∈ On → suc dom 𝐴 ∈ On)
2321, 22syl 17 . . 3 (𝐴 No → suc dom 𝐴 ∈ On)
2420, 23syl5eqel 2734 . 2 (𝐴 No → dom (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ On)
25 rnun 5576 . . . 4 ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩})
26 rnsnopg 5650 . . . . . 6 (dom 𝐴 ∈ V → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
272, 26syl 17 . . . . 5 (𝐴 No → ran {⟨dom 𝐴, 𝑋⟩} = {𝑋})
2827uneq2d 3800 . . . 4 (𝐴 No → (ran 𝐴 ∪ ran {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
2925, 28syl5eq 2697 . . 3 (𝐴 No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) = (ran 𝐴 ∪ {𝑋}))
30 norn 31929 . . . 4 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
31 snssi 4371 . . . . 5 (𝑋 ∈ {1𝑜, 2𝑜} → {𝑋} ⊆ {1𝑜, 2𝑜})
323, 31mp1i 13 . . . 4 (𝐴 No → {𝑋} ⊆ {1𝑜, 2𝑜})
3330, 32unssd 3822 . . 3 (𝐴 No → (ran 𝐴 ∪ {𝑋}) ⊆ {1𝑜, 2𝑜})
3429, 33eqsstrd 3672 . 2 (𝐴 No → ran (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ⊆ {1𝑜, 2𝑜})
35 elno2 31932 . 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 1265 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  {cpr 4212  ⟨cop 4216  dom cdm 5143  ran crn 5144  Ord word 5760  Oncon0 5761  suc csuc 5763  Fun wfun 5920  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-no 31921 This theorem is referenced by:  noextendlt  31947  noextendgt  31948  nosupno  31974  nosupbnd1  31985  nosupbnd2lem1  31986
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