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Theorem n0f 3558
 Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3559 requires only that x not be free in, rather than not occur in, A. (Contributed by NM, 17-Oct-2003.)
Hypothesis
Ref Expression
n0f.1 xA
Assertion
Ref Expression
n0f (Ax x A)

Proof of Theorem n0f
StepHypRef Expression
1 n0f.1 . . . . 5 xA
2 nfcv 2489 . . . . 5 x
31, 2cleqf 2513 . . . 4 (A = x(x Ax ))
4 noel 3554 . . . . . 6 ¬ x
54nbn 336 . . . . 5 x A ↔ (x Ax ))
65albii 1566 . . . 4 (x ¬ x Ax(x Ax ))
73, 6bitr4i 243 . . 3 (A = x ¬ x A)
87necon3abii 2546 . 2 (A ↔ ¬ x ¬ x A)
9 df-ex 1542 . 2 (x x A ↔ ¬ x ¬ x A)
108, 9bitr4i 243 1 (Ax x A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476   ≠ wne 2516  ∅c0 3550 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-dif 3215  df-nul 3551 This theorem is referenced by:  n0  3559  abn0  3568
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