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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.
StepHypRef Expression
1 incom 3448 . . . 4 (R ∩ I ) = ( I ∩ R)
21eqeq1i 2360 . . 3 ((R ∩ I ) = ↔ ( I ∩ R) = )
3 disj5 3890 . . 3 (( I ∩ R) = ↔ I R)
4 ssrel 4844 . . 3 ( I Rxy(x, y I → x, y R))
52, 3, 43bitri 262 . 2 ((R ∩ I ) = xy(x, y I → x, y R))
6 vex 2862 . . . . . 6 y V
76ideq 4870 . . . . 5 (x I yx = y)
8 df-br 4640 . . . . 5 (x I yx, y I )
97, 8bitr3i 242 . . . 4 (x = yx, y I )
10 df-br 4640 . . . . . 6 (xRyx, y R)
1110notbii 287 . . . . 5 xRy ↔ ¬ x, y R)
12 vex 2862 . . . . . . 7 x V
1312, 6opex 4588 . . . . . 6 x, y V
1413elcompl 3225 . . . . 5 (x, y R ↔ ¬ x, y R)
1511, 14bitr4i 243 . . . 4 xRyx, y R)
169, 15imbi12i 316 . . 3 ((x = y → ¬ xRy) ↔ (x, y I → x, y R))
17162albii 1567 . 2 (xy(x = y → ¬ xRy) ↔ xy(x, y I → x, y R))
18 equcom 1680 . . . . . 6 (x = yy = x)
1918imbi1i 315 . . . . 5 ((x = y → ¬ xRy) ↔ (y = x → ¬ xRy))
2019albii 1566 . . . 4 (y(x = y → ¬ xRy) ↔ y(y = x → ¬ xRy))
21 breq2 4643 . . . . . 6 (y = x → (xRyxRx))
2221notbid 285 . . . . 5 (y = x → (¬ xRy ↔ ¬ xRx))
2312, 22ceqsalv 2885 . . . 4 (y(y = x → ¬ xRy) ↔ ¬ xRx)
2420, 23bitri 240 . . 3 (y(x = y → ¬ xRy) ↔ ¬ xRx)
2524albii 1566 . 2 (xy(x = y → ¬ xRy) ↔ x ¬ xRx)
265, 17, 253bitr2i 264 1 ((R ∩ I ) = x ¬ xRx)
 Colors of variables: wff setvar class 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)
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