Theorem csbprc 4350

Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.)

Assertion

Ref	Expression
csbprc	⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Proof of Theorem csbprc

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3739	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
2		falim 1559	. . . 4 ⊢ (⊥ → 𝐴 ∈ V)
3	1, 2	pm5.21ni 377	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
4	3	abbidv 2803	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3839	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		fal 1556	. . . 4 ⊢ ¬ ⊥
7	6	abf 4347	. . 3 ⊢ {𝑦 ∣ ⊥} = ∅
8	7	eqcomi 2746	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
9	4, 5, 8	3eqtr4g 2797	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542  ⊥wfal 1554   ∈ wcel 2114  {cab 2715  Vcvv 3430  [wsbc 3729  ⦋csb 3838  ∅c0 4274

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-nul 4275

This theorem is referenced by:  csb0  4351  sbcel12  4352  sbcne12  4356  sbcel2  4359  csbidm  4374  csbun  4382  csbin  4383  csbdif  4466  csbif  4525  csbuni  4881  sbcbr123  5140  sbcbr  5141  csbexg  5245  csbopab  5501  csbxp  5723  csbres  5939  csbima12  6036  csbrn  6159  csbiota  6483  csbfv12  6877  csbfv  6879  csbriota  7330  csbov123  7402  csbov  7403  csbttc  36712