Theorem elimasni 6094

Description: Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)

Assertion

Ref	Expression
elimasni	⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)

Proof of Theorem elimasni

Step	Hyp	Ref	Expression
1		noel 4299	. . . . 5 ⊢ ¬ 𝐶 ∈ ∅
2		snprc 4688	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3	2	biimpi 219	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅)
4	3	imaeq2d 6063	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅))
5		ima0 6080	. . . . . . 7 ⊢ (𝐴 “ ∅) = ∅
6	4, 5	eqtrdi 2820	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅)
7	6	eleq2d 2855	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅))
8	1, 7	mtbiri 330	. . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵}))
9	8	con4i 115	. . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V)
10		elex 3484	. . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V)
11	9, 10	jca 520	. 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
12		elimasng1 6090	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
13	12	biimpd 232	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶))
14	11, 13	mpcom 39	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {csn 4594   class class class wbr 5113   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  dffv2  6977  poimirlem2  38161  poimirlem23  38182