Theorem intirr 5029

Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Revised 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 3448	. . . 4 ⊢ (R ∩ I ) = ( I ∩ R)
2	1	eqeq1i 2360	. . 3 ⊢ ((R ∩ I ) = ∅ ↔ ( I ∩ R) = ∅)
3		disj5 3890	. . 3 ⊢ (( I ∩ R) = ∅ ↔ I ⊆ ∼ R)
4		ssrel 4844	. . 3 ⊢ ( I ⊆ ∼ R ↔ ∀x∀y(⟨x, y⟩ ∈ I → ⟨x, y⟩ ∈ ∼ R))
5	2, 3, 4	3bitri 262	. 2 ⊢ ((R ∩ I ) = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ I → ⟨x, y⟩ ∈ ∼ R))
6		vex 2862	. . . . . 6 ⊢ y ∈ V
7	6	ideq 4870	. . . . 5 ⊢ (x I y ↔ x = y)
8		df-br 4640	. . . . 5 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
9	7, 8	bitr3i 242	. . . 4 ⊢ (x = y ↔ ⟨x, y⟩ ∈ I )
10		df-br 4640	. . . . . 6 ⊢ (xRy ↔ ⟨x, y⟩ ∈ R)
11	10	notbii 287	. . . . 5 ⊢ (¬ xRy ↔ ¬ ⟨x, y⟩ ∈ R)
12		vex 2862	. . . . . . 7 ⊢ x ∈ V
13	12, 6	opex 4588	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
14	13	elcompl 3225	. . . . 5 ⊢ (⟨x, y⟩ ∈ ∼ R ↔ ¬ ⟨x, y⟩ ∈ R)
15	11, 14	bitr4i 243	. . . 4 ⊢ (¬ xRy ↔ ⟨x, y⟩ ∈ ∼ R)
16	9, 15	imbi12i 316	. . 3 ⊢ ((x = y → ¬ xRy) ↔ (⟨x, y⟩ ∈ I → ⟨x, y⟩ ∈ ∼ R))
17	16	2albii 1567	. 2 ⊢ (∀x∀y(x = y → ¬ xRy) ↔ ∀x∀y(⟨x, y⟩ ∈ I → ⟨x, y⟩ ∈ ∼ R))
18		equcom 1680	. . . . . 6 ⊢ (x = y ↔ y = x)
19	18	imbi1i 315	. . . . 5 ⊢ ((x = y → ¬ xRy) ↔ (y = x → ¬ xRy))
20	19	albii 1566	. . . 4 ⊢ (∀y(x = y → ¬ xRy) ↔ ∀y(y = x → ¬ xRy))
21		breq2 4643	. . . . . 6 ⊢ (y = x → (xRy ↔ xRx))
22	21	notbid 285	. . . . 5 ⊢ (y = x → (¬ xRy ↔ ¬ xRx))
23	12, 22	ceqsalv 2885	. . . 4 ⊢ (∀y(y = x → ¬ xRy) ↔ ¬ xRx)
24	20, 23	bitri 240	. . 3 ⊢ (∀y(x = y → ¬ xRy) ↔ ¬ xRx)
25	24	albii 1566	. 2 ⊢ (∀x∀y(x = y → ¬ xRy) ↔ ∀x ¬ xRx)
26	5, 17, 25	3bitr2i 264	1 ⊢ ((R ∩ I ) = ∅ ↔ ∀x ¬ xRx)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∼ ccompl 3205   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ⟨cop 4561   class class class wbr 4639   I cid 4763

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-id 4767

This theorem is referenced by: (None)