Theorem sltval 27587

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 (∀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 sltval

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

Step	Hyp	Ref	Expression
1		eleq1 2819	. . . . 5 ⊢ (𝑓 = 𝐴 → (𝑓 ∈ No ↔ 𝐴 ∈ No ))
2	1	anbi1d 631	. . . 4 ⊢ (𝑓 = 𝐴 → ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝑔 ∈ No )))
3		fveq1 6821	. . . . . . . 8 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑦) = (𝐴‘𝑦))
4	3	eqeq1d 2733	. . . . . . 7 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝑔‘𝑦)))
5	4	ralbidv 3155	. . . . . 6 ⊢ (𝑓 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦)))
6		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑥) = (𝐴‘𝑥))
7	6	breq1d 5101	. . . . . 6 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))
8	5, 7	anbi12d 632	. . . . 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 3156	. . . 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 632	. . 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 2819	. . . . 5 ⊢ (𝑔 = 𝐵 → (𝑔 ∈ No ↔ 𝐵 ∈ No ))
12	11	anbi2d 630	. . . 4 ⊢ (𝑔 = 𝐵 → ((𝐴 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No )))
13		fveq1 6821	. . . . . . . 8 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑦) = (𝐵‘𝑦))
14	13	eqeq2d 2742	. . . . . . 7 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝐵‘𝑦)))
15	14	ralbidv 3155	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦)))
16		fveq1 6821	. . . . . . 7 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑥) = (𝐵‘𝑥))
17	16	breq2d 5103	. . . . . 6 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
18	15, 17	anbi12d 632	. . . . 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 3156	. . . 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 632	. . 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-slt 27583	. . 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 5479	. 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 541	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ∅c0 4283  {ctp 4580  ⟨cop 4582   class class class wbr 5091  Oncon0 6306  ‘cfv 6481  1_oc1o 8378  2_oc2o 8379   No csur 27579   <s cslt 27580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-iota 6437  df-fv 6489  df-slt 27583

This theorem is referenced by:  sltval2  27596  sltres  27602  nolesgn2o  27611  nogesgn1o  27613  nodense  27632  nolt02o  27635  nogt01o  27636  nosupbnd2lem1  27655  noinfbnd2lem1  27670