Theorem intirr 6150

Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
intirr	⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)

Distinct variable group:   𝑥,𝑅

Proof of Theorem intirr

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		incom 4230	. . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅)
2	1	eqeq1i 2745	. . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3		disj2 4481	. . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4		reli 5850	. . . 4 ⊢ Rel I
5		ssrel 5806	. . . 4 ⊢ (Rel I → ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))))
6	4, 5	ax-mp 5	. . 3 ⊢ ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
7	2, 3, 6	3bitri 297	. 2 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
8		equcom 2017	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
9		vex 3492	. . . . . 6 ⊢ 𝑦 ∈ V
10	9	ideq 5877	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
11		df-br 5167	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
12	8, 10, 11	3bitr2i 299	. . . 4 ⊢ (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13		opex 5484	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
14	13	biantrur 530	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
15		eldif 3986	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
16	14, 15	bitr4i 278	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
17		df-br 5167	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
18	16, 17	xchnxbir 333	. . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
19	12, 18	imbi12i 350	. . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
20	19	2albii 1818	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21		breq2 5170	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥))
22	21	notbid 318	. . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
23	22	equsalvw 2003	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
24	23	albii 1817	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
25	7, 20, 24	3bitr2i 299	1 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654   class class class wbr 5166   I cid 5592  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707

This theorem is referenced by:  hartogslem1  9611  hausdiag  23674