Theorem onnoxpg 44135

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

Assertion

Ref	Expression
onnoxpg	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )

Proof of Theorem onnoxpg

Step	Hyp	Ref	Expression
1		fconst6g 6769	. . 3 ⊢ (𝐵 ∈ {1o, 2o} → (𝐴 × {𝐵}):𝐴⟶{1o, 2o})
2	1	adantl 486	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}):𝐴⟶{1o, 2o})
3		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → (𝐴 × {𝐵}):𝐴⟶{1o, 2o})
4	3	ffund 6712	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → Fun (𝐴 × {𝐵}))
5		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → 𝐵 ∈ {1o, 2o})
6		snnzg 4741	. . . . 5 ⊢ (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅)
7		dmxp 5921	. . . . . 6 ⊢ ({𝐵} ≠ ∅ → dom (𝐴 × {𝐵}) = 𝐴)
8	7	eqcomd 2769	. . . . 5 ⊢ ({𝐵} ≠ ∅ → 𝐴 = dom (𝐴 × {𝐵}))
9	5, 6, 8	3syl 19	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → 𝐴 = dom (𝐴 × {𝐵}))
10		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → 𝐴 ∈ On)
11	9, 10	eqeltrrd 2864	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → dom (𝐴 × {𝐵}) ∈ On)
12	3	frnd 6716	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → ran (𝐴 × {𝐵}) ⊆ {1o, 2o})
13		elno2 27796	. . 3 ⊢ ((𝐴 × {𝐵}) ∈ No ↔ (Fun (𝐴 × {𝐵}) ∧ dom (𝐴 × {𝐵}) ∈ On ∧ ran (𝐴 × {𝐵}) ⊆ {1o, 2o}))
14	4, 11, 12, 13	syl3anbrc 1362	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
15	2, 14	mpd3an3 1491	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ⊆ wss 3906  ∅c0 4287  {csn 4590  {cpr 4592   × cxp 5661  dom cdm 5663  ran crn 5664  Oncon0 6362  Fun wfun 6532  ⟶wf 6534  1_oc1o 8447  2_oc2o 8448   No csur 27782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-fun 6540  df-fn 6541  df-f 6542  df-no 27785

This theorem is referenced by:  onnobdayg  44136  bdaybndex  44137  onnoxp  44139