Theorem bdaybndex 43422

Description: Bounds formed from the birthday are surreal numbers. (Contributed by RP, 21-Sep-2023.)

Assertion

Ref	Expression
bdaybndex	⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )

Proof of Theorem bdaybndex

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = ( bday ‘𝐴))
2		bdayval 27617	. . . . 5 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
3	2	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → ( bday ‘𝐴) = dom 𝐴)
4	1, 3	eqtrd 2771	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 27619	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2835	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 ∈ On)
8		onnog 43420	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
9	7, 8	stoic3 1776	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {csn 4606  {cpr 4608   × cxp 5657  dom cdm 5659  Oncon0 6357  ‘cfv 6536  1_oc1o 8478  2_oc2o 8479   No csur 27608   bday cbday 27610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-no 27611  df-bday 27613

This theorem is referenced by:  bdaybndbday  43423