Theorem elno3 27567

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 1096	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
2		elno2 27566	. 2 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
3		df-f 6515	. . . 4 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
4		funfn 6546	. . . . 5 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
5	4	anbi1i 624	. . . 4 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
6	3, 5	bitr4i 278	. . 3 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
7	6	anbi1i 624	. 2 ⊢ ((𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
8	1, 2, 7	3bitr4i 303	1 ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1o, 2o} ∧ dom 𝐴 ∈ On))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   ⊆ wss 3914  {cpr 4591  dom cdm 5638  ran crn 5639  Oncon0 6332  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  1_oc1o 8427  2_oc2o 8428   No csur 27551

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-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-fun 6513  df-fn 6514  df-f 6515  df-no 27554

This theorem is referenced by:  noxp1o  27575  noseponlem  27576