Theorem onnobdayg 44234

Description: Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.)

Assertion

Ref	Expression
onnobdayg	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴)

Proof of Theorem onnobdayg

Step	Hyp	Ref	Expression
1		onnoxpg 44233	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
2		bdayval 27868	. . 3 ⊢ ((𝐴 × {𝐵}) ∈ No → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵}))
3	1, 2	syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵}))
4		simpr 490	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → 𝐵 ∈ {1o, 2o})
5		snnzg 4742	. . 3 ⊢ (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅)
6		dmxp 5921	. . 3 ⊢ ({𝐵} ≠ ∅ → dom (𝐴 × {𝐵}) = 𝐴)
7	4, 5, 6	3syl 19	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → dom (𝐴 × {𝐵}) = 𝐴)
8	3, 7	eqtrd 2800	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∅c0 4286  {csn 4591  {cpr 4593   × cxp 5661  dom cdm 5663  Oncon0 6365  ‘cfv 6541  1_oc1o 8453  2_oc2o 8454   No csur 27860   bday cbday 27862

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7743

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-no 27863  df-bday 27865

This theorem is used by: (None)