Theorem bdaybndex 44578

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 490	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → 𝐵 = (bday‘𝐴))
2		bdayval 28020	. . . . 5 ⊢ (𝐴 ∈ No → (bday‘𝐴) = dom 𝐴)
3	2	adantr 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → (bday‘𝐴) = dom 𝐴)
4	1, 3	eqtrd 2798	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 28022	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 486	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2864	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → 𝐵 ∈ On)
8		onnoxpg 44576	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No)
9	7, 8	stoic3 1808	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  {csn 4588  {cpr 4590   × cxp 5653  dom cdm 5655  Oncon0 6369  ‘cfv 6547  1_oc1o 8483  2_oc2o 8484  Nocsur 28012  bdaycbday 28014

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-no 28015  df-bday 28017

This theorem is used by:  bdaybndbday  44579