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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

Proof of Theorem nnsucelrlem2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldifsn 3839 . . . . 5
2 elun 3220 . . . . . . 7
3 elsn 3748 . . . . . . . 8
43orbi2i 505 . . . . . . 7
52, 4bitri 240 . . . . . 6
6 df-ne 2518 . . . . . 6
75, 6anbi12i 678 . . . . 5
8 pm5.61 693 . . . . 5
91, 7, 83bitri 262 . . . 4
10 ancom 437 . . . 4
119, 10bitri 240 . . 3
12 eleq1 2413 . . . . . . . 8
1312biimpcd 215 . . . . . . 7
1413con3d 125 . . . . . 6
1514com12 27 . . . . 5
1615pm4.71rd 616 . . . 4
1716bicomd 192 . . 3
1811, 17syl5bb 248 . 2
1918eqrdv 2351 1
 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-mp 5  ax-1 6  ax-2 7  ax-3 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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