Theorem int0 2542

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44.

Assertion

Ref	Expression
int0	⊢ ∩∅ = V

Proof of Theorem int0

Step	Hyp	Ref	Expression
1		noel 2280	. . . . . 6 ⊢ ¬ y ∈ ∅
2	1	pm2.21i 77	. . . . 5 ⊢ (y ∈ ∅ → x ∈ y)
3	2	ax-gen 961	. . . 4 ⊢ ∀y(y ∈ ∅ → x ∈ y)
4		eqid 1473	. . . 4 ⊢ x = x
5	3, 4	2th 717	. . 3 ⊢ (∀y(y ∈ ∅ → x ∈ y) ↔ x = x)
6	5	abbii 1572	. 2 ⊢ {x∣∀y(y ∈ ∅ → x ∈ y)} = {x∣x = x}
7		df-int 2529	. 2 ⊢ ∩∅ = {x∣∀y(y ∈ ∅ → x ∈ y)}
8		df-v 1808	. 2 ⊢ V = {x∣x = x}
9	6, 7, 8	3eqtr4 1502	1 ⊢ ∩∅ = V

Syntax hints:   → wi 3  ∀wal 952   = wceq 954   ∈ wcel 956  {cab 1461  Vcvv 1807  ∅c0 2276  ∩cint 2528

This theorem is referenced by:  intex 2724  intnex 2725  oev2 4152  fiint 4540  fiiu 10386  fiiu2 10413  efilcp 10481  efilcp2 10486  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-dif 2045  df-nul 2277  df-int 2529

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