Theorem csbprc 4013

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 3478	. . . 4 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V)
2		falim 1538	. . . 4 ⊢ (⊥ → 𝐴 ∈ V)
3	1, 2	pm5.21ni 366	. . 3 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥))
4	3	abbidv 2770	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥})
5		df-csb 3567	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
6		fal 1530	. . . 4 ⊢ ¬ ⊥
7	6	abf 4011	. . 3 ⊢ {𝑦 ∣ ⊥} = ∅
8	7	eqcomi 2660	. 2 ⊢ ∅ = {𝑦 ∣ ⊥}
9	4, 5, 8	3eqtr4g 2710	1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523  ⊥wfal 1528   ∈ wcel 2030  {cab 2637  Vcvv 3231  [wsbc 3468  ⦋csb 3566  ∅c0 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949

This theorem is referenced by:  csb0  4015  sbcel12  4016  sbcne12  4019  sbcel2  4022  csbidm  4035  csbun  4042  csbin  4043  csbif  4171  csbuni  4498  sbcbr123  4739  sbcbr  4740  csbexg  4825  csbopab  5037  csbxp  5234  csbres  5431  csbima12  5518  csbrn  5631  csbiota  5919  csbfv12  6269  csbfv  6271  csbriota  6663  csbov123  6727  csbov  6728  csbdif  33301