Theorem epinid0 9546

Description: The membership relation and the identity relation are disjoint. Variable-free version of nelaneq 9543. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.)

Assertion

Ref	Expression
epinid0	⊢ ( E ∩ I ) = ∅

Proof of Theorem epinid0

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

Step	Hyp	Ref	Expression
1		df-eprel 5543	. . 3 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
2		df-id 5538	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
3	1, 2	ineq12i 4168	. 2 ⊢ ( E ∩ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
4		inopab 5798	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)}
5		nelaneq 9543	. . . 4 ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)
6	5	gen2 1815	. . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)
7		opab0 5521	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦))
8	6, 7	mpbir 233	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} = ∅
9	3, 4, 8	3eqtri 2788	1 ⊢ ( E ∩ I ) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1557   = wceq 1559   ∩ cin 3901  ∅c0 4283  {copab 5159   I cid 5537   E cep 5542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-reg 9533

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650

This theorem is referenced by: (None)