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Theorem uniintsn 3963
 Description: Two ways to express "A is a singleton." See also en1 in set.mm, en1b in set.mm, card1 in set.mm, and eusn 3796. (Contributed by NM, 2-Aug-2010.)
Assertion
Ref Expression
uniintsn (A = Ax A = {x})
Distinct variable group:   x,A

Proof of Theorem uniintsn
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 vn0 3557 . . . . . 6 V ≠
2 inteq 3929 . . . . . . . . . . 11 (A = A = )
3 int0 3940 . . . . . . . . . . 11 = V
42, 3syl6eq 2401 . . . . . . . . . 10 (A = A = V)
54adantl 452 . . . . . . . . 9 ((A = A A = ) → A = V)
6 unieq 3900 . . . . . . . . . . . 12 (A = A = )
7 uni0 3918 . . . . . . . . . . . 12 =
86, 7syl6eq 2401 . . . . . . . . . . 11 (A = A = )
9 eqeq1 2359 . . . . . . . . . . 11 (A = A → (A = A = ))
108, 9syl5ib 210 . . . . . . . . . 10 (A = A → (A = A = ))
1110imp 418 . . . . . . . . 9 ((A = A A = ) → A = )
125, 11eqtr3d 2387 . . . . . . . 8 ((A = A A = ) → V = )
1312ex 423 . . . . . . 7 (A = A → (A = → V = ))
1413necon3d 2554 . . . . . 6 (A = A → (V ≠ A))
151, 14mpi 16 . . . . 5 (A = AA)
16 n0 3559 . . . . 5 (Ax x A)
1715, 16sylib 188 . . . 4 (A = Ax x A)
18 vex 2862 . . . . . . 7 x V
19 vex 2862 . . . . . . 7 y V
2018, 19prss 3861 . . . . . 6 ((x A y A) ↔ {x, y} A)
21 uniss 3912 . . . . . . . . . . . . 13 ({x, y} A{x, y} A)
2221adantl 452 . . . . . . . . . . . 12 ((A = A {x, y} A) → {x, y} A)
23 simpl 443 . . . . . . . . . . . 12 ((A = A {x, y} A) → A = A)
2422, 23sseqtrd 3307 . . . . . . . . . . 11 ((A = A {x, y} A) → {x, y} A)
25 intss 3947 . . . . . . . . . . . 12 ({x, y} AA {x, y})
2625adantl 452 . . . . . . . . . . 11 ((A = A {x, y} A) → A {x, y})
2724, 26sstrd 3282 . . . . . . . . . 10 ((A = A {x, y} A) → {x, y} {x, y})
2818, 19unipr 3905 . . . . . . . . . 10 {x, y} = (xy)
2918, 19intpr 3959 . . . . . . . . . 10 {x, y} = (xy)
3027, 28, 293sstr3g 3311 . . . . . . . . 9 ((A = A {x, y} A) → (xy) (xy))
31 inss1 3475 . . . . . . . . . 10 (xy) x
32 ssun1 3426 . . . . . . . . . 10 x (xy)
3331, 32sstri 3281 . . . . . . . . 9 (xy) (xy)
3430, 33jctir 524 . . . . . . . 8 ((A = A {x, y} A) → ((xy) (xy) (xy) (xy)))
35 eqss 3287 . . . . . . . . 9 ((xy) = (xy) ↔ ((xy) (xy) (xy) (xy)))
36 uneqin 3506 . . . . . . . . 9 ((xy) = (xy) ↔ x = y)
3735, 36bitr3i 242 . . . . . . . 8 (((xy) (xy) (xy) (xy)) ↔ x = y)
3834, 37sylib 188 . . . . . . 7 ((A = A {x, y} A) → x = y)
3938ex 423 . . . . . 6 (A = A → ({x, y} Ax = y))
4020, 39syl5bi 208 . . . . 5 (A = A → ((x A y A) → x = y))
4140alrimivv 1632 . . . 4 (A = Axy((x A y A) → x = y))
4217, 41jca 518 . . 3 (A = A → (x x A xy((x A y A) → x = y)))
43 euabsn 3792 . . . 4 (∃!x x Ax{x x A} = {x})
44 eleq1 2413 . . . . 5 (x = y → (x Ay A))
4544eu4 2243 . . . 4 (∃!x x A ↔ (x x A xy((x A y A) → x = y)))
46 abid2 2470 . . . . . 6 {x x A} = A
4746eqeq1i 2360 . . . . 5 ({x x A} = {x} ↔ A = {x})
4847exbii 1582 . . . 4 (x{x x A} = {x} ↔ x A = {x})
4943, 45, 483bitr3i 266 . . 3 ((x x A xy((x A y A) → x = y)) ↔ x A = {x})
5042, 49sylib 188 . 2 (A = Ax A = {x})
5118unisn 3907 . . . 4 {x} = x
52 unieq 3900 . . . 4 (A = {x} → A = {x})
53 inteq 3929 . . . . 5 (A = {x} → A = {x})
5418intsn 3962 . . . . 5 {x} = x
5553, 54syl6eq 2401 . . . 4 (A = {x} → A = x)
5651, 52, 553eqtr4a 2411 . . 3 (A = {x} → A = A)
5756exlimiv 1634 . 2 (x A = {x} → A = A)
5850, 57impbii 180 1 (A = Ax A = {x})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339   ≠ wne 2516  Vcvv 2859   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  {csn 3737  {cpr 3738  ∪cuni 3891  ∩cint 3926 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 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-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-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-pr 3742  df-uni 3892  df-int 3927 This theorem is referenced by:  uniintab  3964
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