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Theorem sltval 27150
Description: The value of the surreal less-than relation. (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 2822 . . . . 5 (𝑓 = 𝐴 → (𝑓 No 𝐴 No ))
21anbi1d 631 . . . 4 (𝑓 = 𝐴 → ((𝑓 No 𝑔 No ) ↔ (𝐴 No 𝑔 No )))
3 fveq1 6891 . . . . . . . 8 (𝑓 = 𝐴 → (𝑓𝑦) = (𝐴𝑦))
43eqeq1d 2735 . . . . . . 7 (𝑓 = 𝐴 → ((𝑓𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝑔𝑦)))
54ralbidv 3178 . . . . . 6 (𝑓 = 𝐴 → (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦)))
6 fveq1 6891 . . . . . . 7 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
76breq1d 5159 . . . . . 6 (𝑓 = 𝐴 → ((𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)))
85, 7anbi12d 632 . . . . 5 (𝑓 = 𝐴 → ((∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥))))
98rexbidv 3179 . . . 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 632 . . 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 2822 . . . . 5 (𝑔 = 𝐵 → (𝑔 No 𝐵 No ))
1211anbi2d 630 . . . 4 (𝑔 = 𝐵 → ((𝐴 No 𝑔 No ) ↔ (𝐴 No 𝐵 No )))
13 fveq1 6891 . . . . . . . 8 (𝑔 = 𝐵 → (𝑔𝑦) = (𝐵𝑦))
1413eqeq2d 2744 . . . . . . 7 (𝑔 = 𝐵 → ((𝐴𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
1514ralbidv 3178 . . . . . 6 (𝑔 = 𝐵 → (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)))
16 fveq1 6891 . . . . . . 7 (𝑔 = 𝐵 → (𝑔𝑥) = (𝐵𝑥))
1716breq2d 5161 . . . . . 6 (𝑔 = 𝐵 → ((𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))
1815, 17anbi12d 632 . . . . 5 (𝑔 = 𝐵 → ((∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥))))
1918rexbidv 3179 . . . 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 632 . . 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 27147 . . 3 <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔𝑥)))}
2210, 20, 21brabg 5540 . 2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵𝑥)))))
2322bianabs 543 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  c0 4323  {ctp 4633  cop 4635   class class class wbr 5149  Oncon0 6365  cfv 6544  1oc1o 8459  2oc2o 8460   No csur 27143   <s cslt 27144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-slt 27147
This theorem is referenced by:  sltval2  27159  sltres  27165  nolesgn2o  27174  nogesgn1o  27176  nodense  27195  nolt02o  27198  nogt01o  27199  nosupbnd2lem1  27218  noinfbnd2lem1  27233
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