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Theorem noextendlt 31947
Description: Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
Assertion
Ref Expression
noextendlt (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)

Proof of Theorem noextendlt
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nofun 31927 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
2 funfn 5956 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
31, 2sylib 208 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
4 nodmon 31928 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
5 1on 7612 . . . . . . . . 9 1𝑜 ∈ On
6 fnsng 5976 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
74, 5, 6sylancl 695 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
8 nodmord 31931 . . . . . . . . . 10 (𝐴 No → Ord dom 𝐴)
9 ordirr 5779 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
108, 9syl 17 . . . . . . . . 9 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
11 disjsn 4278 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1210, 11sylibr 224 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13 snidg 4239 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
144, 13syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
15 fvun2 6309 . . . . . . . 8 ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
163, 7, 12, 14, 15syl112anc 1370 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
17 fvsng 6488 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
184, 5, 17sylancl 695 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
1916, 18eqtrd 2685 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜)
20 ndmfv 6256 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
2110, 20syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
2219, 21jca 553 . . . . 5 (𝐴 No → (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅))
23223mix1d 1256 . . . 4 (𝐴 No → ((((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = 2𝑜) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ∅ ∧ (𝐴‘dom 𝐴) = 2𝑜)))
24 fvex 6239 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) ∈ V
25 fvex 6239 . . . . 5 (𝐴‘dom 𝐴) ∈ V
2624, 25brtp 31765 . . . 4 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘dom 𝐴) ↔ ((((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = 2𝑜) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ∅ ∧ (𝐴‘dom 𝐴) = 2𝑜)))
2723, 26sylibr 224 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘dom 𝐴))
28 necom 2876 . . . . . . 7 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥))
2928rabbii 3216 . . . . . 6 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
3029inteqi 4511 . . . . 5 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
315elexi 3244 . . . . . . 7 1𝑜 ∈ V
3231prid1 4329 . . . . . 6 1𝑜 ∈ {1𝑜, 2𝑜}
3332noextenddif 31946 . . . . 5 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)} = dom 𝐴)
3430, 33syl5eq 2697 . . . 4 (𝐴 No {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = dom 𝐴)
3534fveq2d 6233 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴))
3634fveq2d 6233 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}) = (𝐴‘dom 𝐴))
3727, 35, 363brtr4d 4717 . 2 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}))
3832noextend 31944 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No )
39 sltval2 31934 . . 3 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No 𝐴 No ) → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴 ↔ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)})))
4038, 39mpancom 704 . 2 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴 ↔ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)})))
4137, 40mpbird 247 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1053   = wceq 1523  wcel 2030  wne 2823  {crab 2945  cun 3605  cin 3606  c0 3948  {csn 4210  {ctp 4214  cop 4216   cint 4507   class class class wbr 4685  dom cdm 5143  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918   <s cslt 31919
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-pow 4873  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-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  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-1o 7605  df-2o 7606  df-no 31921  df-slt 31922
This theorem is referenced by: (None)
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