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 27711	. . . . 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 2780	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 27713	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2844	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 ∈ On)
8		onnog 43391	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
9	7, 8	stoic3 1774	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {csn 4648  {cpr 4650   × cxp 5698  dom cdm 5700  Oncon0 6395  ‘cfv 6573  1_oc1o 8515  2_oc2o 8516   No csur 27702   bday cbday 27704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-no 27705  df-bday 27707

This theorem is referenced by:  bdaybndbday  43394