Theorem elno3 27721

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 1109	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
2		elno2 27720	. 2 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
3		df-f 6527	. . . 4 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
4		funfn 6553	. . . . 5 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
5	4	anbi1i 633	. . . 4 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
6	3, 5	bitr4i 280	. . 3 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
7	6	anbi1i 633	. 2 ⊢ ((𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
8	1, 2, 7	3bitr4i 305	1 ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   ∈ wcel 2144   ⊆ wss 3906  {cpr 4586  dom cdm 5649  ran crn 5650  Oncon0 6348  Fun wfun 6517   Fn wfn 6518  ⟶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-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-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  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-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-fun 6525  df-fn 6526  df-f 6527  df-no 27709

This theorem is referenced by:  noxp1o  27729  noseponlem  27730