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Theorem sltval 33051
Description: The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
sltval ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem sltval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2897 . . . . 5 (𝑓 = 𝐴 → (𝑓 No 𝐴 No ))
21anbi1d 629 . . . 4 (𝑓 = 𝐴 → ((𝑓 No 𝑔 No ) ↔ (𝐴 No 𝑔 No )))
3 fveq1 6662 . . . . . . . 8 (𝑓 = 𝐴 → (𝑓𝑦) = (𝐴𝑦))
43eqeq1d 2820 . . . . . . 7 (𝑓 = 𝐴 → ((𝑓𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝑔𝑦)))
54ralbidv 3194 . . . . . 6 (𝑓 = 𝐴 → (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦)))
6 fveq1 6662 . . . . . . 7 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
76breq1d 5067 . . . . . 6 (𝑓 = 𝐴 → ((𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)))
85, 7anbi12d 630 . . . . 5 (𝑓 = 𝐴 → ((∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥))))
98rexbidv 3294 . . . 4 (𝑓 = 𝐴 → (∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)) ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥))))
102, 9anbi12d 630 . . 3 (𝑓 = 𝐴 → (((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥))) ↔ ((𝐴 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)))))
11 eleq1 2897 . . . . 5 (𝑔 = 𝐵 → (𝑔 No 𝐵 No ))
1211anbi2d 628 . . . 4 (𝑔 = 𝐵 → ((𝐴 No 𝑔 No ) ↔ (𝐴 No 𝐵 No )))
13 fveq1 6662 . . . . . . . 8 (𝑔 = 𝐵 → (𝑔𝑦) = (𝐵𝑦))
1413eqeq2d 2829 . . . . . . 7 (𝑔 = 𝐵 → ((𝐴𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
1514ralbidv 3194 . . . . . 6 (𝑔 = 𝐵 → (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)))
16 fveq1 6662 . . . . . . 7 (𝑔 = 𝐵 → (𝑔𝑥) = (𝐵𝑥))
1716breq2d 5069 . . . . . 6 (𝑔 = 𝐵 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
1815, 17anbi12d 630 . . . . 5 (𝑔 = 𝐵 → ((∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
1918rexbidv 3294 . . . 4 (𝑔 = 𝐵 → (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)) ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
2012, 19anbi12d 630 . . 3 (𝑔 = 𝐵 → (((𝐴 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥))) ↔ ((𝐴 No 𝐵 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))))
21 df-slt 33048 . . 3 <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)))}
2210, 20, 21brabg 5417 . 2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))))
2322bianabs 542 1 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  c0 4288  {ctp 4561  cop 4563   class class class wbr 5057  Oncon0 6184  cfv 6348  1oc1o 8084  2oc2o 8085   No csur 33044   <s cslt 33045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356  df-slt 33048
This theorem is referenced by:  sltval2  33060  sltres  33066  nolesgn2o  33075  nodense  33093  nolt02o  33096  nosupbnd2lem1  33112
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