Theorem elimasni 6051

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 4279	. . . . 5 ⊢ ¬ 𝐶 ∈ ∅
2		snprc 4662	. . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3	2	biimpi 216	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅)
4	3	imaeq2d 6020	. . . . . . 7 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = (𝐴 “ ∅))
5		ima0 6037	. . . . . . 7 ⊢ (𝐴 “ ∅) = ∅
6	4, 5	eqtrdi 2788	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → (𝐴 “ {𝐵}) = ∅)
7	6	eleq2d 2823	. . . . 5 ⊢ (¬ 𝐵 ∈ V → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ ∅))
8	1, 7	mtbiri 327	. . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ 𝐶 ∈ (𝐴 “ {𝐵}))
9	8	con4i 114	. . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵 ∈ V)
10		elex 3451	. . 3 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐶 ∈ V)
11	9, 10	jca 511	. 2 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → (𝐵 ∈ V ∧ 𝐶 ∈ V))
12		elimasng1 6047	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
13	12	biimpd 229	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶))
14	11, 13	mpcom 38	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) → 𝐵𝐴𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  {csn 4568   class class class wbr 5086   “ cima 5628

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-pr 5371

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-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  dffv2  6930  poimirlem2  37960  poimirlem23  37981