Theorem onnobdayg 43406

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		onnog 43405	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
2		bdayval 27576	. . 3 ⊢ ((𝐴 × {𝐵}) ∈ No → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵}))
3	1, 2	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵}))
4		simpr 484	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → 𝐵 ∈ {1o, 2o})
5		snnzg 4728	. . 3 ⊢ (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅)
6		dmxp 5875	. . 3 ⊢ ({𝐵} ≠ ∅ → dom (𝐴 × {𝐵}) = 𝐴)
7	4, 5, 6	3syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → dom (𝐴 × {𝐵}) = 𝐴)
8	3, 7	eqtrd 2764	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∅c0 4286  {csn 4579  {cpr 4581   × cxp 5621  dom cdm 5623  Oncon0 6311  ‘cfv 6486  1_oc1o 8388  2_oc2o 8389   No csur 27567   bday cbday 27569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-no 27570  df-bday 27572

This theorem is referenced by: (None)