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Theorem pw10b 4166
 Description: The unit power class of a class is empty iff the class itself is empty. (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
pw10b (1A = A = )

Proof of Theorem pw10b
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 n0 3559 . . . 4 (Ax x A)
2 snelpw1 4146 . . . . . 6 ({x} 1Ax A)
3 ne0i 3556 . . . . . 6 ({x} 1A1A)
42, 3sylbir 204 . . . . 5 (x A1A)
54exlimiv 1634 . . . 4 (x x A1A)
61, 5sylbi 187 . . 3 (A1A)
76necon4i 2576 . 2 (1A = A = )
8 pw1eq 4143 . . 3 (A = 1A = 1)
9 pw10 4161 . . 3 1 =
108, 9syl6eq 2401 . 2 (A = 1A = )
117, 10impbii 180 1 (1A = A = )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∅c0 3550  {csn 3737  ℘1cpw1 4135 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-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-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137 This theorem is referenced by:  ncfinlower  4483
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