Theorem bdaybndex 44431

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 28005	. . . . 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 2796	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → 𝐵 = dom 𝐴)
5		nodmon 28007	. . . 4 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
6	5	adantr 486	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → dom 𝐴 ∈ On)
7	4, 6	eqeltrd 2861	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 = (bday‘𝐴)) → 𝐵 ∈ On)
8		onnoxpg 44429	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No)
9	7, 8	stoic3 1809	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 1570   ∈ wcel 2145  {csn 4584  {cpr 4586   × cxp 5649  dom cdm 5651  Oncon0 6362  ‘cfv 6538  1_oc1o 8469  2_oc2o 8470  Nocsur 27997  bdaycbday 27999

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-no 28000  df-bday 28002

This theorem is used by:  bdaybndbday  44432