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Theorem noextendlt 33513
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 33493 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
2 funfn 6369 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
31, 2sylib 221 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
4 nodmon 33494 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
5 1on 8138 . . . . . . . . 9 1o ∈ On
6 fnsng 6391 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 1o ∈ On) → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
74, 5, 6sylancl 589 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
8 nodmord 33497 . . . . . . . . . 10 (𝐴 No → Ord dom 𝐴)
9 ordirr 6190 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
108, 9syl 17 . . . . . . . . 9 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
11 disjsn 4602 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1210, 11sylibr 237 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13 snidg 4550 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
144, 13syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
15 fvun2 6760 . . . . . . . 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 1375 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
17 fvsng 6952 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 1o ∈ On) → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
184, 5, 17sylancl 589 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
1916, 18eqtrd 2773 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o)
20 ndmfv 6704 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
2110, 20syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
2219, 21jca 515 . . . . 5 (𝐴 No → (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅))
23223mix1d 1337 . . . 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 6687 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) ∈ V
25 fvex 6687 . . . . 5 (𝐴‘dom 𝐴) ∈ V
2624, 25brtp 33288 . . . 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 237 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘dom 𝐴))
28 necom 2987 . . . . . . 7 (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥))
2928rabbii 3374 . . . . . 6 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
3029inteqi 4840 . . . . 5 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
31 1oex 8144 . . . . . . 7 1o ∈ V
3231prid1 4653 . . . . . 6 1o ∈ {1o, 2o}
3332noextenddif 33512 . . . . 5 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)} = dom 𝐴)
3430, 33syl5eq 2785 . . . 4 (𝐴 No {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = dom 𝐴)
3534fveq2d 6678 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴))
3634fveq2d 6678 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)}) = (𝐴‘dom 𝐴))
3727, 35, 363brtr4d 5062 . 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 33510 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No )
39 sltval2 33500 . . 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 688 . 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 260 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3o 1087   = wceq 1542  wcel 2114  wne 2934  {crab 3057  cun 3841  cin 3842  c0 4211  {csn 4516  {ctp 4520  cop 4522   cint 4836   class class class wbr 5030  dom cdm 5525  Ord word 6171  Oncon0 6172  Fun wfun 6333   Fn wfn 6334  cfv 6339  1oc1o 8124  2oc2o 8125   No csur 33484   <s cslt 33485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-1o 8131  df-2o 8132  df-no 33487  df-slt 33488
This theorem is referenced by:  noinfbnd1  33573
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