Theorem bdaybndex 43393

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 27536	. . . . 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 2764	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 27538	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2828	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 ∈ On)
8		onnog 43391	. 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 4585  {cpr 4587   × cxp 5629  dom cdm 5631  Oncon0 6320  ‘cfv 6499  1_oc1o 8404  2_oc2o 8405   No csur 27527   bday cbday 27529

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 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-no 27530  df-bday 27532

This theorem is referenced by:  bdaybndbday  43394