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Theorem el1st 4729
 Description: Membership in 1st. (Contributed by SF, 5-Jan-2015.)
Assertion
Ref Expression
el1st (A 1stxy A = x, y, x)
Distinct variable group:   x,A,y

Proof of Theorem el1st
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-1st 4723 . . . 4 1st = {z, x y z = x, y}
21eleq2i 2417 . . 3 (A 1stA {z, x y z = x, y})
3 elopab 4696 . . 3 (A {z, x y z = x, y} ↔ zx(A = z, x y z = x, y))
42, 3bitri 240 . 2 (A 1stzx(A = z, x y z = x, y))
5 excom 1741 . . 3 (zx(A = z, x y z = x, y) ↔ xz(A = z, x y z = x, y))
6 excom 1741 . . . . 5 (yz(A = z, x z = x, y) ↔ zy(A = z, x z = x, y))
7 exancom 1586 . . . . . . 7 (z(A = z, x z = x, y) ↔ z(z = x, y A = z, x))
8 vex 2862 . . . . . . . . 9 x V
9 vex 2862 . . . . . . . . 9 y V
108, 9opex 4588 . . . . . . . 8 x, y V
11 opeq1 4578 . . . . . . . . 9 (z = x, yz, x = x, y, x)
1211eqeq2d 2364 . . . . . . . 8 (z = x, y → (A = z, xA = x, y, x))
1310, 12ceqsexv 2894 . . . . . . 7 (z(z = x, y A = z, x) ↔ A = x, y, x)
147, 13bitri 240 . . . . . 6 (z(A = z, x z = x, y) ↔ A = x, y, x)
1514exbii 1582 . . . . 5 (yz(A = z, x z = x, y) ↔ y A = x, y, x)
16 exdistr 1906 . . . . 5 (zy(A = z, x z = x, y) ↔ z(A = z, x y z = x, y))
176, 15, 163bitr3ri 267 . . . 4 (z(A = z, x y z = x, y) ↔ y A = x, y, x)
1817exbii 1582 . . 3 (xz(A = z, x y z = x, y) ↔ xy A = x, y, x)
195, 18bitri 240 . 2 (zx(A = z, x y z = x, y) ↔ xy A = x, y, x)
204, 19bitri 240 1 (A 1stxy A = x, y, x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622  1st c1st 4717 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-1st 4723 This theorem is referenced by:  br1stg  4730
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