Theorem predprc 6341

Description: The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.)

Assertion

Ref	Expression
predprc	⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)

Proof of Theorem predprc

Step	Hyp	Ref	Expression
1		df-pred 6304	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		snprc 4684	. . . . . . 7 ⊢ (¬ 𝑋 ∈ V ↔ {𝑋} = ∅)
3	2	biimpi 219	. . . . . 6 ⊢ (¬ 𝑋 ∈ V → {𝑋} = ∅)
4	3	imaeq2d 6064	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (◡𝑅 “ {𝑋}) = (◡𝑅 “ ∅))
5		ima0 6081	. . . . 5 ⊢ (◡𝑅 “ ∅) = ∅
6	4, 5	eqtrdi 2814	. . . 4 ⊢ (¬ 𝑋 ∈ V → (◡𝑅 “ {𝑋}) = ∅)
7	6	ineq2d 4174	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ ∅))
8		in0 4353	. . 3 ⊢ (𝐴 ∩ ∅) = ∅
9	7, 8	eqtrdi 2814	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐴 ∩ (◡𝑅 “ {𝑋})) = ∅)
10	1, 9	eqtrid 2810	1 ⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3905  ∅c0 4287  {csn 4590  ^◡ccnv 5662   “ cima 5666  Predcpred 6303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304

This theorem is referenced by:  predres  6342