Theorem ltsval 27769

Description: The value of the surreal less-than relation. (Contributed by Scott Fenton, 14-Jun-2011.)

Assertion

Ref	Expression
ltsval	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ltsval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2853	. . . . 5 ⊢ (𝑓 = 𝐴 → (𝑓 ∈ No ↔ 𝐴 ∈ No ))
2	1	anbi1d 642	. . . 4 ⊢ (𝑓 = 𝐴 → ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝑔 ∈ No )))
3		fveq1 6870	. . . . . . . 8 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑦) = (𝐴‘𝑦))
4	3	eqeq1d 2767	. . . . . . 7 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝑔‘𝑦)))
5	4	ralbidv 3188	. . . . . 6 ⊢ (𝑓 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦)))
6		fveq1 6870	. . . . . . 7 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑥) = (𝐴‘𝑥))
7	6	breq1d 5115	. . . . . 6 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))
8	5, 7	anbi12d 643	. . . . 5 ⊢ (𝑓 = 𝐴 → ((∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)) ↔ (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥))))
9	8	rexbidv 3189	. . . 4 ⊢ (𝑓 = 𝐴 → (∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)) ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥))))
10	2, 9	anbi12d 643	. . 3 ⊢ (𝑓 = 𝐴 → (((𝑓 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥))) ↔ ((𝐴 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))))
11		eleq1 2853	. . . . 5 ⊢ (𝑔 = 𝐵 → (𝑔 ∈ No ↔ 𝐵 ∈ No ))
12	11	anbi2d 641	. . . 4 ⊢ (𝑔 = 𝐵 → ((𝐴 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No )))
13		fveq1 6870	. . . . . . . 8 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑦) = (𝐵‘𝑦))
14	13	eqeq2d 2776	. . . . . . 7 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝐵‘𝑦)))
15	14	ralbidv 3188	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦)))
16		fveq1 6870	. . . . . . 7 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑥) = (𝐵‘𝑥))
17	16	breq2d 5117	. . . . . 6 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
18	15, 17	anbi12d 643	. . . . 5 ⊢ (𝑔 = 𝐵 → ((∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)) ↔ (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
19	18	rexbidv 3189	. . . 4 ⊢ (𝑔 = 𝐵 → (∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)) ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
20	12, 19	anbi12d 643	. . 3 ⊢ (𝑔 = 𝐵 → (((𝐴 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥))) ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))))
21		df-lts 27766	. . 3 ⊢ <s = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))}
22	10, 20, 21	brabg 5515	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))))
23	22	bianabs 550	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ∅c0 4288  {ctp 4589  ⟨cop 4591   class class class wbr 5105  Oncon0 6350  ‘cfv 6525  1_oc1o 8434  2_oc2o 8435   No csur 27762   <s clts 27763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-iota 6481  df-fv 6533  df-lts 27766

This theorem is referenced by:  ltsval2  27778  ltsres  27784  nolesgn2o  27793  nogesgn1o  27795  nodense  27814  nolt02o  27817  nogt01o  27818  nosupbnd2lem1  27837  noinfbnd2lem1  27852