Theorem ltsval 27986

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 2849	. . . . 5 ⊢ (𝑓 = 𝐴 → (𝑓 ∈ No ↔ 𝐴 ∈ No ))
2	1	anbi1d 643	. . . 4 ⊢ (𝑓 = 𝐴 → ((𝑓 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝑔 ∈ No )))
3		fveq1 6876	. . . . . . . 8 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑦) = (𝐴‘𝑦))
4	3	eqeq1d 2763	. . . . . . 7 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝑔‘𝑦)))
5	4	ralbidv 3186	. . . . . 6 ⊢ (𝑓 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦)))
6		fveq1 6876	. . . . . . 7 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑥) = (𝐴‘𝑥))
7	6	breq1d 5113	. . . . . 6 ⊢ (𝑓 = 𝐴 → ((𝑓‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥)))
8	5, 7	anbi12d 644	. . . . 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 3187	. . . 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 644	. . 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 2849	. . . . 5 ⊢ (𝑔 = 𝐵 → (𝑔 ∈ No ↔ 𝐵 ∈ No ))
12	11	anbi2d 642	. . . 4 ⊢ (𝑔 = 𝐵 → ((𝐴 ∈ No ∧ 𝑔 ∈ No ) ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No )))
13		fveq1 6876	. . . . . . . 8 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑦) = (𝐵‘𝑦))
14	13	eqeq2d 2772	. . . . . . 7 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑦) = (𝑔‘𝑦) ↔ (𝐴‘𝑦) = (𝐵‘𝑦)))
15	14	ralbidv 3186	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝑔‘𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦)))
16		fveq1 6876	. . . . . . 7 ⊢ (𝑔 = 𝐵 → (𝑔‘𝑥) = (𝐵‘𝑥))
17	16	breq2d 5115	. . . . . 6 ⊢ (𝑔 = 𝐵 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝑔‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
18	15, 17	anbi12d 644	. . . . 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 3187	. . . 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 644	. . 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 27983	. . 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 5514	. 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 551	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  ∅c0 4279  {ctp 4588  ⟨cop 4590   class class class wbr 5103  Oncon0 6355  ‘cfv 6531  1_oc1o 8453  2_oc2o 8454   No csur 27979   <s clts 27980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6487  df-fv 6539  df-lts 27983

This theorem is used by:  ltsval2  27995  ltsres  28001  nolesgn2o  28010  nogesgn1o  28012  nodense  28031  nolt02o  28034  nogt01o  28035  nosupbnd2lem1  28054  noinfbnd2lem1  28069