Theorem epinid0 9498

Description: The membership relation and the identity relation are disjoint. Variable-free version of nelaneq 9496. (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 5521	. . 3 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
2		df-id 5516	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
3	1, 2	ineq12i 4167	. 2 ⊢ ( E ∩ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
4		inopab 5775	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)}
5		nelaneq 9496	. . . 4 ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)
6	5	gen2 1797	. . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)
7		opab0 5499	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦))
8	6, 7	mpbir 231	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} = ∅
9	3, 4, 8	3eqtri 2760	1 ⊢ ( E ∩ I ) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1539   = wceq 1541   ∩ cin 3897  ∅c0 4282  {copab 5157   I cid 5515   E cep 5520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-reg 9487

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5158  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628

This theorem is referenced by: (None)