Theorem intasym 5028

Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 9-Sep-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
intasym	⊢ ((R ∩ ◡R) ⊆ I ↔ ∀x∀y((xRy ∧ yRx) → x = y))

Distinct variable group:   x,y,R

Proof of Theorem intasym

Step	Hyp	Ref	Expression
1		ssrel 4844	. 2 ⊢ ((R ∩ ◡R) ⊆ I ↔ ∀x∀y(⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I ))
2		df-br 4640	. . . . 5 ⊢ (x(R ∩ ◡R)y ↔ ⟨x, y⟩ ∈ (R ∩ ◡R))
3		brin 4693	. . . . . 6 ⊢ (x(R ∩ ◡R)y ↔ (xRy ∧ x◡Ry))
4		brcnv 4892	. . . . . . 7 ⊢ (x◡Ry ↔ yRx)
5	4	anbi2i 675	. . . . . 6 ⊢ ((xRy ∧ x◡Ry) ↔ (xRy ∧ yRx))
6	3, 5	bitri 240	. . . . 5 ⊢ (x(R ∩ ◡R)y ↔ (xRy ∧ yRx))
7	2, 6	bitr3i 242	. . . 4 ⊢ (⟨x, y⟩ ∈ (R ∩ ◡R) ↔ (xRy ∧ yRx))
8		df-br 4640	. . . . 5 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
9		vex 2862	. . . . . 6 ⊢ y ∈ V
10	9	ideq 4870	. . . . 5 ⊢ (x I y ↔ x = y)
11	8, 10	bitr3i 242	. . . 4 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
12	7, 11	imbi12i 316	. . 3 ⊢ ((⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I ) ↔ ((xRy ∧ yRx) → x = y))
13	12	2albii 1567	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I ) ↔ ∀x∀y((xRy ∧ yRx) → x = y))
14	1, 13	bitri 240	1 ⊢ ((R ∩ ◡R) ⊆ I ↔ ∀x∀y((xRy ∧ yRx) → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257  ⟨cop 4561   class class class wbr 4639   I cid 4763  ^◡ccnv 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  df-cnv 4785

This theorem is referenced by: (None)