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Theorem sltval 31784
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 2688 . . . . 5 (𝑓 = 𝐴 → (𝑓 No 𝐴 No ))
21anbi1d 741 . . . 4 (𝑓 = 𝐴 → ((𝑓 No 𝑔 No ) ↔ (𝐴 No 𝑔 No )))
3 fveq1 6188 . . . . . . . 8 (𝑓 = 𝐴 → (𝑓𝑦) = (𝐴𝑦))
43eqeq1d 2623 . . . . . . 7 (𝑓 = 𝐴 → ((𝑓𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝑔𝑦)))
54ralbidv 2985 . . . . . 6 (𝑓 = 𝐴 → (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦)))
6 fveq1 6188 . . . . . . 7 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
76breq1d 4661 . . . . . 6 (𝑓 = 𝐴 → ((𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))
85, 7anbi12d 747 . . . . 5 (𝑓 = 𝐴 → ((∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥))))
98rexbidv 3050 . . . 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 747 . . 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 2688 . . . . 5 (𝑔 = 𝐵 → (𝑔 No 𝐵 No ))
1211anbi2d 740 . . . 4 (𝑔 = 𝐵 → ((𝐴 No 𝑔 No ) ↔ (𝐴 No 𝐵 No )))
13 fveq1 6188 . . . . . . . 8 (𝑔 = 𝐵 → (𝑔𝑦) = (𝐵𝑦))
1413eqeq2d 2631 . . . . . . 7 (𝑔 = 𝐵 → ((𝐴𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
1514ralbidv 2985 . . . . . 6 (𝑔 = 𝐵 → (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)))
16 fveq1 6188 . . . . . . 7 (𝑔 = 𝐵 → (𝑔𝑥) = (𝐵𝑥))
1716breq2d 4663 . . . . . 6 (𝑔 = 𝐵 → ((𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
1815, 17anbi12d 747 . . . . 5 (𝑔 = 𝐵 → ((∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
1918rexbidv 3050 . . . 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 747 . . 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 31781 . . 3 <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))}
2210, 20, 21brabg 4992 . 2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))))
2322bianabs 924 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 196  wa 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912  c0 3913  {ctp 4179  cop 4181   class class class wbr 4651  Oncon0 5721  cfv 5886  1𝑜c1o 7550  2𝑜c2o 7551   No csur 31777   <s cslt 31778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-iota 5849  df-fv 5894  df-slt 31781
This theorem is referenced by:  sltval2  31793  sltres  31799  nolesgn2o  31808  nodense  31826  nolt02o  31829  nosupbnd2lem1  31845
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