Theorem elno3 27637

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 1097	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) ∧ dom 𝐴 ∈ On))
2		elno2 27636	. 2 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
3		df-f 6498	. . . 4 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
4		funfn 6524	. . . . 5 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
5	4	anbi1i 625	. . . 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 625	. 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 1087   ∈ wcel 2114   ⊆ wss 3890  {cpr 4570  dom cdm 5626  ran crn 5627  Oncon0 6319  Fun wfun 6488   Fn wfn 6489  ⟶wf 6490  1_oc1o 8393  2_oc2o 8394   No csur 27621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-fun 6496  df-fn 6497  df-f 6498  df-no 27624

This theorem is referenced by:  noxp1o  27645  noseponlem  27646