Theorem int0OLD 4426

Description: Obsolete proof of int0 4425 as of 26-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
int0OLD	⊢ ∩ ∅ = V

Proof of Theorem int0OLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3878	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	pm2.21i 115	. . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
3	2	ax-gen 1713	. . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
4		equid 1926	. . . 4 ⊢ 𝑥 = 𝑥
5	3, 4	2th 253	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥)
6	5	abbii 2726	. 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥}
7		df-int 4411	. 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)}
8		df-v 3175	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2642	1 ⊢ ∩ ∅ = V

Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  ∅c0 3874  ∩ cint 4410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875  df-int 4411

This theorem is referenced by: (None)