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Theorem noextendgt 33873
Description: Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
Assertion
Ref Expression
noextendgt (𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))

Proof of Theorem noextendgt
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nodmord 33856 . . . . . . . 8 (𝐴 No → Ord dom 𝐴)
2 ordirr 6284 . . . . . . . 8 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
31, 2syl 17 . . . . . . 7 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
4 ndmfv 6804 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
53, 4syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
6 nofun 33852 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
7 funfn 6464 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
86, 7sylib 217 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
9 nodmon 33853 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
10 2on 8311 . . . . . . . . 9 2o ∈ On
11 fnsng 6486 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 2o ∈ On) → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
129, 10, 11sylancl 586 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
13 disjsn 4647 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
143, 13sylibr 233 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15 snidg 4595 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
169, 15syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
17 fvun2 6860 . . . . . . . 8 ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
188, 12, 14, 16, 17syl112anc 1373 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
19 fvsng 7052 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 2o ∈ On) → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
209, 10, 19sylancl 586 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
2118, 20eqtrd 2778 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)
225, 21jca 512 . . . . 5 (𝐴 No → ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o))
23223mix3d 1337 . . . 4 (𝐴 No → (((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ∅) ∨ ((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o) ∨ ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)))
24 fvex 6787 . . . . 5 (𝐴‘dom 𝐴) ∈ V
25 fvex 6787 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ∈ V
2624, 25brtp 33717 . . . 4 ((𝐴‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ↔ (((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ∅) ∨ ((𝐴‘dom 𝐴) = 1o ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o) ∨ ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)))
2723, 26sylibr 233 . . 3 (𝐴 No → (𝐴‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴))
2810elexi 3451 . . . . . 6 2o ∈ V
2928prid2 4699 . . . . 5 2o ∈ {1o, 2o}
3029noextenddif 33871 . . . 4 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)} = dom 𝐴)
3130fveq2d 6778 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
3230fveq2d 6778 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴))
3327, 31, 323brtr4d 5106 . 2 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}))
3429noextend 33869 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No )
35 sltval2 33859 . . 3 ((𝐴 No ∧ (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No ) → (𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)})))
3634, 35mpdan 684 . 2 (𝐴 No → (𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)})))
3733, 36mpbird 256 1 (𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3o 1085   = wceq 1539  wcel 2106  wne 2943  {crab 3068  cun 3885  cin 3886  c0 4256  {csn 4561  {ctp 4565  cop 4567   cint 4879   class class class wbr 5074  dom cdm 5589  Ord word 6265  Oncon0 6266  Fun wfun 6427   Fn wfn 6428  cfv 6433  1oc1o 8290  2oc2o 8291   No csur 33843   <s cslt 33844
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847
This theorem is referenced by:  nosupbnd1  33917
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