Theorem sn-tz6.12-2 42670

Description: tz6.12-2 6869 without ax-10 2142, ax-11 2158, ax-12 2178. 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 5128	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑦))
2		breq2 5128	. . . 4 ⊢ (𝑥 = 𝑧 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝑧))
3	1, 2	euabsn2w 42669	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦})
4	3	notbii 320	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦})
5		df-fv 6544	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
6		iotanul2 6506	. . 3 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (℩𝑥𝐴𝐹𝑥) = ∅)
7	5, 6	eqtrid 2783	. 2 ⊢ (¬ ∃𝑦{𝑥 ∣ 𝐴𝐹𝑥} = {𝑦} → (𝐹‘𝐴) = ∅)
8	4, 7	sylbi 217	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  ∃wex 1779  ∃!weu 2568  {cab 2714  ∅c0 4313  {csn 4606   class class class wbr 5124  ℩cio 6487  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544

This theorem is referenced by: (None)