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Theorem sltval 31501
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 2686 . . . . 5 (𝑓 = 𝐴 → (𝑓 No 𝐴 No ))
21anbi1d 740 . . . 4 (𝑓 = 𝐴 → ((𝑓 No 𝑔 No ) ↔ (𝐴 No 𝑔 No )))
3 fveq1 6147 . . . . . . . 8 (𝑓 = 𝐴 → (𝑓𝑦) = (𝐴𝑦))
43eqeq1d 2623 . . . . . . 7 (𝑓 = 𝐴 → ((𝑓𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝑔𝑦)))
54ralbidv 2980 . . . . . 6 (𝑓 = 𝐴 → (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦)))
6 fveq1 6147 . . . . . . 7 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
76breq1d 4623 . . . . . 6 (𝑓 = 𝐴 → ((𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))
85, 7anbi12d 746 . . . . 5 (𝑓 = 𝐴 → ((∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥))))
98rexbidv 3045 . . . 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 746 . . 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 2686 . . . . 5 (𝑔 = 𝐵 → (𝑔 No 𝐵 No ))
1211anbi2d 739 . . . 4 (𝑔 = 𝐵 → ((𝐴 No 𝑔 No ) ↔ (𝐴 No 𝐵 No )))
13 fveq1 6147 . . . . . . . 8 (𝑔 = 𝐵 → (𝑔𝑦) = (𝐵𝑦))
1413eqeq2d 2631 . . . . . . 7 (𝑔 = 𝐵 → ((𝐴𝑦) = (𝑔𝑦) ↔ (𝐴𝑦) = (𝐵𝑦)))
1514ralbidv 2980 . . . . . 6 (𝑔 = 𝐵 → (∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ↔ ∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦)))
16 fveq1 6147 . . . . . . 7 (𝑔 = 𝐵 → (𝑔𝑥) = (𝐵𝑥))
1716breq2d 4625 . . . . . 6 (𝑔 = 𝐵 → ((𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥) ↔ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))
1815, 17anbi12d 746 . . . . 5 (𝑔 = 𝐵 → ((∀𝑦𝑥 (𝐴𝑦) = (𝑔𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)) ↔ (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥))))
1918rexbidv 3045 . . . 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 746 . . 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 31498 . . 3 <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 No 𝑔 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝑓𝑦) = (𝑔𝑦) ∧ (𝑓𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑔𝑥)))}
2210, 20, 21brabg 4954 . 2 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 ↔ ((𝐴 No 𝐵 No ) ∧ ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴𝑦) = (𝐵𝑦) ∧ (𝐴𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵𝑥)))))
2322bianabs 923 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 1480  wcel 1987  wral 2907  wrex 2908  c0 3891  {ctp 4152  cop 4154   class class class wbr 4613  Oncon0 5682  cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499   No csur 31494   <s cslt 31495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-iota 5810  df-fv 5855  df-slt 31498
This theorem is referenced by:  sltval2  31510  sltres  31518  nodense  31552  nobndup  31563  nobnddown  31564
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