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Theorem eluni1g 4172
 Description: Membership in a unit union. (Contributed by SF, 15-Mar-2015.)
Assertion
Ref Expression
eluni1g 1

Proof of Theorem eluni1g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-uni1 4138 . . . 4 1 1c
21eleq2i 2417 . . 3 1 1c
3 eluni 3894 . . 3 1c 1c
4 elin 3219 . . . . . . . 8 1c 1c
5 ancom 437 . . . . . . . 8 1c 1c
6 el1c 4139 . . . . . . . . . 10 1c
76anbi1i 676 . . . . . . . . 9 1c
8 19.41v 1901 . . . . . . . . 9
97, 8bitr4i 243 . . . . . . . 8 1c
104, 5, 93bitri 262 . . . . . . 7 1c
1110anbi2i 675 . . . . . 6 1c
12 19.42v 1905 . . . . . 6
1311, 12bitr4i 243 . . . . 5 1c
1413exbii 1582 . . . 4 1c
15 excom 1741 . . . 4
16 an12 772 . . . . . . 7
1716exbii 1582 . . . . . 6
18 snex 4111 . . . . . . 7
19 eleq2 2414 . . . . . . . . 9
20 vex 2862 . . . . . . . . . 10
2120elsnc2 3762 . . . . . . . . 9
2219, 21syl6bb 252 . . . . . . . 8
23 eleq1 2413 . . . . . . . 8
2422, 23anbi12d 691 . . . . . . 7
2518, 24ceqsexv 2894 . . . . . 6
26 eqcom 2355 . . . . . . 7
2726anbi1i 676 . . . . . 6
2817, 25, 273bitri 262 . . . . 5
2928exbii 1582 . . . 4
3014, 15, 293bitri 262 . . 3 1c
312, 3, 303bitri 262 . 2 1
32 sneq 3744 . . . 4
3332eleq1d 2419 . . 3
3433ceqsexgv 2971 . 2
3531, 34syl5bb 248 1 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wex 1541   wceq 1642   wcel 1710   cin 3208  csn 3737  cuni 3891  ⋃1cuni1 4133  1cc1c 4134 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-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892  df-1c 4136  df-uni1 4138 This theorem is referenced by:  eluni1  4173
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