Theorem bdaybndex 43421

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 27708	. . . . 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 2775	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 27710	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2839	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 ∈ On)
8		onnog 43419	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
9	7, 8	stoic3 1773	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {csn 4631  {cpr 4633   × cxp 5687  dom cdm 5689  Oncon0 6386  ‘cfv 6563  1_oc1o 8498  2_oc2o 8499   No csur 27699   bday cbday 27701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-no 27702  df-bday 27704

This theorem is referenced by:  bdaybndbday  43422