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Theorem noextendlt 32334
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 32314 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
2 funfn 6132 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
31, 2sylib 210 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
4 nodmon 32315 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
5 1on 7807 . . . . . . . . 9 1𝑜 ∈ On
6 fnsng 6153 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
74, 5, 6sylancl 581 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
8 nodmord 32318 . . . . . . . . . 10 (𝐴 No → Ord dom 𝐴)
9 ordirr 5960 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
108, 9syl 17 . . . . . . . . 9 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
11 disjsn 4437 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1210, 11sylibr 226 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13 snidg 4399 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
144, 13syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
15 fvun2 6496 . . . . . . . 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 1494 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
17 fvsng 6677 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
184, 5, 17sylancl 581 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
1916, 18eqtrd 2834 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜)
20 ndmfv 6442 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
2110, 20syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
2219, 21jca 508 . . . . 5 (𝐴 No → (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅))
23223mix1d 1436 . . . 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 6425 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) ∈ V
25 fvex 6425 . . . . 5 (𝐴‘dom 𝐴) ∈ V
2624, 25brtp 32152 . . . 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 226 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘dom 𝐴))
28 necom 3025 . . . . . . 7 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥))
2928rabbii 3370 . . . . . 6 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
3029inteqi 4672 . . . . 5 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
31 1oex 7808 . . . . . . 7 1𝑜 ∈ V
3231prid1 4487 . . . . . 6 1𝑜 ∈ {1𝑜, 2𝑜}
3332noextenddif 32333 . . . . 5 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)} = dom 𝐴)
3430, 33syl5eq 2846 . . . 4 (𝐴 No {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = dom 𝐴)
3534fveq2d 6416 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴))
3634fveq2d 6416 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}) = (𝐴‘dom 𝐴))
3727, 35, 363brtr4d 4876 . 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 32331 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No )
39 sltval2 32321 . . 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 680 . 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 249 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3o 1107   = wceq 1653  wcel 2157  wne 2972  {crab 3094  cun 3768  cin 3769  c0 4116  {csn 4369  {ctp 4373  cop 4375   cint 4668   class class class wbr 4844  dom cdm 5313  Ord word 5941  Oncon0 5942  Fun wfun 6096   Fn wfn 6097  cfv 6102  1𝑜c1o 7793  2𝑜c2o 7794   No csur 32305   <s cslt 32306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-ord 5945  df-on 5946  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-1o 7800  df-2o 7801  df-no 32308  df-slt 32309
This theorem is referenced by: (None)
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