Theorem bdaybndex 44007

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 488	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = ( bday ‘𝐴))
2		bdayval 27712	. . . . 5 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)
3	2	adantr 484	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → ( bday ‘𝐴) = dom 𝐴)
4	1, 3	eqtrd 2797	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 27714	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 484	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2862	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴)) → 𝐵 ∈ On)
8		onnoxpg 44005	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
9	7, 8	stoic3 1796	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = ( bday ‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  {csn 4582  {cpr 4584   × cxp 5645  dom cdm 5647  Oncon0 6346  ‘cfv 6521  1_oc1o 8430  2_oc2o 8431   No csur 27704   bday cbday 27706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-no 27707  df-bday 27709

This theorem is referenced by:  bdaybndbday  44008