Theorem noxp1o 27729

Description: The Cartesian product of an ordinal and {1_o} is a surreal. (Contributed by Scott Fenton, 12-Jun-2011.)

Assertion

Ref	Expression
noxp1o	⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No )

Proof of Theorem noxp1o

Step	Hyp	Ref	Expression
1		1oex 8449	. . . . . 6 ⊢ 1o ∈ V
2	1	prid1 4723	. . . . 5 ⊢ 1o ∈ {1o, 2o}
3	2	fconst6 6756	. . . 4 ⊢ (𝐴 × {1o}):𝐴⟶{1o, 2o}
4	1	snnz 4737	. . . . . 6 ⊢ {1o} ≠ ∅
5		dmxp 5907	. . . . . 6 ⊢ ({1o} ≠ ∅ → dom (𝐴 × {1o}) = 𝐴)
6	4, 5	ax-mp 5	. . . . 5 ⊢ dom (𝐴 × {1o}) = 𝐴
7	6	feq2i 6685	. . . 4 ⊢ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ↔ (𝐴 × {1o}):𝐴⟶{1o, 2o})
8	3, 7	mpbir 233	. . 3 ⊢ (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o}
9	8	a1i 11	. 2 ⊢ (𝐴 ∈ On → (𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o})
10	6	eleq1i 2855	. . 3 ⊢ (dom (𝐴 × {1o}) ∈ On ↔ 𝐴 ∈ On)
11	10	biimpri 230	. 2 ⊢ (𝐴 ∈ On → dom (𝐴 × {1o}) ∈ On)
12		elno3 27721	. 2 ⊢ ((𝐴 × {1o}) ∈ No ↔ ((𝐴 × {1o}):dom (𝐴 × {1o})⟶{1o, 2o} ∧ dom (𝐴 × {1o}) ∈ On))
13	9, 11, 12	sylanbrc 592	1 ⊢ (𝐴 ∈ On → (𝐴 × {1o}) ∈ No )

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∅c0 4287  {csn 4584  {cpr 4586   × cxp 5647  dom cdm 5649  Oncon0 6348  ⟶wf 6519  1_oc1o 8432  2_oc2o 8433   No csur 27706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-suc 6354  df-fun 6525  df-fn 6526  df-f 6527  df-1o 8439  df-no 27709

This theorem is referenced by:  bdayfo  27743