Theorem elno3 28012

Description: Another condition for membership in No. (Contributed by Scott Fenton, 14-Apr-2012.)

Assertion

Ref	Expression
elno3	⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On))

Proof of Theorem elno3

Step	Hyp	Ref	Expression
1		3anan32 1113	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
2		elno2 28011	. 2 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
3		df-f 6542	. . . 4 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
4		funfn 6570	. . . . 5 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
5	4	anbi1i 636	. . . 4 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
6	3, 5	bitr4i 281	. . 3 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
7	6	anbi1i 636	. 2 ⊢ ((𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
8	1, 2, 7	3bitr4i 306	1 ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144   ⊆ wss 3895  {cpr 4585  dom cdm 5650  ran crn 5651  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  1_oc1o 8469  2_oc2o 8470  Nocsur 27997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-fun 6540  df-fn 6541  df-f 6542  df-no 28000

This theorem is used by:  noxp1o  28020  noseponlem  28021