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Theorem sn-tz6.12-2 42690
Description: tz6.12-2 6894 without ax-10 2141, ax-11 2157, ax-12 2177. Improves 118 theorems. (Contributed by SN, 27-May-2025.)
Assertion
Ref Expression
sn-tz6.12-2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
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.
StepHypRef Expression
1 breq2 5147 . . . 4 (𝑥 = 𝑦 → (𝐴𝐹𝑥𝐴𝐹𝑦))
2 breq2 5147 . . . 4 (𝑥 = 𝑧 → (𝐴𝐹𝑥𝐴𝐹𝑧))
31, 2euabsn2w 42689 . . 3 (∃!𝑥 𝐴𝐹𝑥 ↔ ∃𝑦{𝑥𝐴𝐹𝑥} = {𝑦})
43notbii 320 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ ¬ ∃𝑦{𝑥𝐴𝐹𝑥} = {𝑦})
5 df-fv 6569 . . 3 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
6 iotanul2 6531 . . 3 (¬ ∃𝑦{𝑥𝐴𝐹𝑥} = {𝑦} → (℩𝑥𝐴𝐹𝑥) = ∅)
75, 6eqtrid 2789 . 2 (¬ ∃𝑦{𝑥𝐴𝐹𝑥} = {𝑦} → (𝐹𝐴) = ∅)
84, 7sylbi 217 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wex 1779  ∃!weu 2568  {cab 2714  c0 4333  {csn 4626   class class class wbr 5143  cio 6512  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by: (None)
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