Theorem tz6.12-2OLD 6867

Description: Obsolete version of tz6.12-2 6866 as of 25-Jan-2026. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tz6.12-2OLD	⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz6.12-2OLD

Step	Hyp	Ref	Expression
1		df-fv 6541	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		iotanul 6513	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
3	1, 2	eqtrid 2807	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃!weu 2593  ∅c0 4279   class class class wbr 5103  ℩cio 6487  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3902  df-ss 3916  df-nul 4280  df-sn 4585  df-uni 4868  df-iota 6489  df-fv 6541

This theorem is used by: (None)