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Theorem noextendlt 33060
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 33040 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
2 funfn 6382 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
31, 2sylib 219 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
4 nodmon 33041 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
5 1on 8100 . . . . . . . . 9 1o ∈ On
6 fnsng 6403 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 1o ∈ On) → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
74, 5, 6sylancl 586 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
8 nodmord 33044 . . . . . . . . . 10 (𝐴 No → Ord dom 𝐴)
9 ordirr 6207 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
108, 9syl 17 . . . . . . . . 9 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
11 disjsn 4646 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1210, 11sylibr 235 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13 snidg 4596 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
144, 13syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
15 fvun2 6752 . . . . . . . 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 1368 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
17 fvsng 6938 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 1o ∈ On) → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
184, 5, 17sylancl 586 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
1916, 18eqtrd 2861 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o)
20 ndmfv 6697 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
2110, 20syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
2219, 21jca 512 . . . . 5 (𝐴 No → (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅))
23223mix1d 1330 . . . 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 6680 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) ∈ V
25 fvex 6680 . . . . 5 (𝐴‘dom 𝐴) ∈ V
2624, 25brtp 32869 . . . 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 235 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘dom 𝐴))
28 necom 3074 . . . . . . 7 (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥))
2928rabbii 3479 . . . . . 6 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
3029inteqi 4878 . . . . 5 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
31 1oex 8101 . . . . . . 7 1o ∈ V
3231prid1 4697 . . . . . 6 1o ∈ {1o, 2o}
3332noextenddif 33059 . . . . 5 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)} = dom 𝐴)
3430, 33syl5eq 2873 . . . 4 (𝐴 No {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = dom 𝐴)
3534fveq2d 6671 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴))
3634fveq2d 6671 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)}) = (𝐴‘dom 𝐴))
3727, 35, 363brtr4d 5095 . 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 33057 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No )
39 sltval2 33047 . . 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 684 . 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 258 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3o 1080   = wceq 1530  wcel 2107  wne 3021  {crab 3147  cun 3938  cin 3939  c0 4295  {csn 4564  {ctp 4568  cop 4570   cint 4874   class class class wbr 5063  dom cdm 5554  Ord word 6188  Oncon0 6189  Fun wfun 6346   Fn wfn 6347  cfv 6352  1oc1o 8086  2oc2o 8087   No csur 33031   <s cslt 33032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-1o 8093  df-2o 8094  df-no 33034  df-slt 33035
This theorem is referenced by: (None)
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