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Theorem nnsucelrlem2 4425
 Description: Lemma for nnsucelr 4428. Subtracting a non-element from a set adjoined with the non-element retrieves the original set. (Contributed by SF, 15-Jan-2015.)
Assertion
Ref Expression
nnsucelrlem2 B A → ((A ∪ {B}) {B}) = A)

Proof of Theorem nnsucelrlem2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eldifsn 3839 . . . . 5 (x ((A ∪ {B}) {B}) ↔ (x (A ∪ {B}) xB))
2 elun 3220 . . . . . . 7 (x (A ∪ {B}) ↔ (x A x {B}))
3 elsn 3748 . . . . . . . 8 (x {B} ↔ x = B)
43orbi2i 505 . . . . . . 7 ((x A x {B}) ↔ (x A x = B))
52, 4bitri 240 . . . . . 6 (x (A ∪ {B}) ↔ (x A x = B))
6 df-ne 2518 . . . . . 6 (xB ↔ ¬ x = B)
75, 6anbi12i 678 . . . . 5 ((x (A ∪ {B}) xB) ↔ ((x A x = B) ¬ x = B))
8 pm5.61 693 . . . . 5 (((x A x = B) ¬ x = B) ↔ (x A ¬ x = B))
91, 7, 83bitri 262 . . . 4 (x ((A ∪ {B}) {B}) ↔ (x A ¬ x = B))
10 ancom 437 . . . 4 ((x A ¬ x = B) ↔ (¬ x = B x A))
119, 10bitri 240 . . 3 (x ((A ∪ {B}) {B}) ↔ (¬ x = B x A))
12 eleq1 2413 . . . . . . . 8 (x = B → (x AB A))
1312biimpcd 215 . . . . . . 7 (x A → (x = BB A))
1413con3d 125 . . . . . 6 (x A → (¬ B A → ¬ x = B))
1514com12 27 . . . . 5 B A → (x A → ¬ x = B))
1615pm4.71rd 616 . . . 4 B A → (x A ↔ (¬ x = B x A)))
1716bicomd 192 . . 3 B A → ((¬ x = B x A) ↔ x A))
1811, 17syl5bb 248 . 2 B A → (x ((A ∪ {B}) {B}) ↔ x A))
1918eqrdv 2351 1 B A → ((A ∪ {B}) {B}) = A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206   ∪ cun 3207  {csn 3737 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-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-sn 3741 This theorem is referenced by:  nnsucelr  4428  enadj  6060
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