Theorem sltval 26995

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 2825	. . . . 5 ⊢ (𝑓 = 𝐴 → (𝑓 ∈ No ↔ 𝐴 ∈ No ))
2	1	anbi1d 630	. . . 4 ⊢ (𝑓 = 𝐴 → ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝑔 ∈ No )))
3		fveq1 6841	. . . . . . . 8 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑦) = (𝐴‘𝑦))
4	3	eqeq1d 2738	. . . . . . 7 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝑔‘𝑦)))
5	4	ralbidv 3174	. . . . . 6 ⊢ (𝑓 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦)))
6		fveq1 6841	. . . . . . 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 631	. . . . 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 3175	. . . 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 631	. . 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 2825	. . . . 5 ⊢ (𝑔 = 𝐵 → (𝑔 ∈ No ↔ 𝐵 ∈ No ))
12	11	anbi2d 629	. . . 4 ⊢ (𝑔 = 𝐵 → ((𝐴 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No )))
13		fveq1 6841	. . . . . . . 8 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑦) = (𝐵‘𝑦))
14	13	eqeq2d 2747	. . . . . . 7 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝐵‘𝑦)))
15	14	ralbidv 3174	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦)))
16		fveq1 6841	. . . . . . 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 631	. . . . 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 3175	. . . 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 631	. . 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 26992	. . 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 5496	. 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 542	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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  ∅c0 4282  {ctp 4590  ⟨cop 4592   class class class wbr 5105  Oncon0 6317  ‘cfv 6496  1_oc1o 8405  2_oc2o 8406   No csur 26988   <s cslt 26989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-iota 6448  df-fv 6504  df-slt 26992

This theorem is referenced by:  sltval2  27004  sltres  27010  nolesgn2o  27019  nogesgn1o  27021  nodense  27040  nolt02o  27043  nogt01o  27044  nosupbnd2lem1  27063  noinfbnd2lem1  27078