Theorem int0 3629

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

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

Step	Hyp	Ref	Expression
1		noel 3228	. . . . . 6 ⊢ ¬ 𝑦 ∈ ∅
2	1	pm2.21i 575	. . . . 5 ⊢ (𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
3	2	ax-gen 1338	. . . 4 ⊢ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)
4		equid 1589	. . . 4 ⊢ 𝑥 = 𝑥
5	3, 4	2th 163	. . 3 ⊢ (∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦) ↔ 𝑥 = 𝑥)
6	5	abbii 2153	. 2 ⊢ {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)} = {𝑥 ∣ 𝑥 = 𝑥}
7		df-int 3616	. 2 ⊢ ∩ ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥 ∈ 𝑦)}
8		df-v 2559	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9	6, 7, 8	3eqtr4i 2070	1 ⊢ ∩ ∅ = V

Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  Vcvv 2557  ∅c0 3224  ∩ cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-int 3616

This theorem is referenced by:  rint0  3654  intexr  3904  bj-intexr  10028