Theorem sn-tz6.12-2 42772

Description: tz6.12-2 6809 without ax-10 2144, ax-11 2160, ax-12 2180. 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 5093	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦))
2		breq2 5093	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑧))
3	1, 2	euabsn2w 42771	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦})
4	3	notbii 320	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦})
5		df-fv 6489	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
6		iotanul2 6454	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (℩𝑥𝐴𝐹𝑥) = ∅)
7	5, 6	eqtrid 2778	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (𝐹‘𝐴) = ∅)
8	4, 7	sylbi 217	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541  ∃wex 1780  ∃!weu 2563  {cab 2709  ∅c0 4280  {csn 4573   class class class wbr 5089  ℩cio 6435  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489

This theorem is referenced by: (None)