Theorem sn-tz6.12-2 42667

Description: tz6.12-2 6895 without ax-10 2139, ax-11 2155, ax-12 2175. Improves 118 theorems. (Contributed by SN, 27-May-2025.)

Assertion

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

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem sn-tz6.12-2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5152	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦))
2		breq2 5152	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑧))
3	1, 2	euabsn2w 42666	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦})
4	3	notbii 320	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦})
5		df-fv 6571	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
6		iotanul2 6533	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (℩𝑥𝐴𝐹𝑥) = ∅)
7	5, 6	eqtrid 2787	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (𝐹‘𝐴) = ∅)
8	4, 7	sylbi 217	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537  ∃wex 1776  ∃!weu 2566  {cab 2712  ∅c0 4339  {csn 4631   class class class wbr 5148  ℩cio 6514  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571

This theorem is referenced by: (None)