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Theorem noextendgt 27173
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 27156 . . . . . . . 8 (𝐴 No → Ord dom 𝐴)
2 ordirr 6383 . . . . . . . 8 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
31, 2syl 17 . . . . . . 7 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
4 ndmfv 6927 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
53, 4syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
6 nofun 27152 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
7 funfn 6579 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
86, 7sylib 217 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
9 nodmon 27153 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
10 2on 8480 . . . . . . . . 9 2o ∈ On
11 fnsng 6601 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 2o ∈ On) → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
129, 10, 11sylancl 587 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 2o⟩} Fn {dom 𝐴})
13 disjsn 4716 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
143, 13sylibr 233 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15 snidg 4663 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
169, 15syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
17 fvun2 6984 . . . . . . . 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 1375 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2o⟩}‘dom 𝐴))
19 fvsng 7178 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 2o ∈ On) → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
209, 10, 19sylancl 587 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 2o⟩}‘dom 𝐴) = 2o)
2118, 20eqtrd 2773 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o)
225, 21jca 513 . . . . 5 (𝐴 No → ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) = 2o))
23223mix3d 1339 . . . 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 6905 . . . . 5 (𝐴‘dom 𝐴) ∈ V
25 fvex 6905 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴) ∈ V
2624, 25brtp 5524 . . . 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 3494 . . . . . 6 2o ∈ V
2928prid2 4768 . . . . 5 2o ∈ {1o, 2o}
3029noextenddif 27171 . . . 4 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)} = dom 𝐴)
3130fveq2d 6896 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
3230fveq2d 6896 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 2o⟩})‘dom 𝐴))
3327, 31, 323brtr4d 5181 . 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 27169 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 2o⟩}) ∈ No )
35 sltval2 27159 . . 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 686 . 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 257 1 (𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2o⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3o 1087   = wceq 1542  wcel 2107  wne 2941  {crab 3433  cun 3947  cin 3948  c0 4323  {csn 4629  {ctp 4633  cop 4635   cint 4951   class class class wbr 5149  dom cdm 5677  Ord word 6364  Oncon0 6365  Fun wfun 6538   Fn wfn 6539  cfv 6544  1oc1o 8459  2oc2o 8460   No csur 27143   <s cslt 27144
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 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147
This theorem is referenced by:  nosupbnd1  27217
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