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Theorem iotanul2 6531
Description: Version of iotanul 6539 using df-iota 6514 instead of dfiota2 6515. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
iotanul2 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotanul2
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iota 6514 . 2 (℩𝑥𝜑) = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
2 n0 4353 . . . 4 ( {𝑤 ∣ {𝑥𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}})
3 eluni 4910 . . . . . 6 (𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} ↔ ∃𝑦(𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}))
4 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
5 sneq 4636 . . . . . . . . . . 11 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2748 . . . . . . . . . 10 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6elab 3679 . . . . . . . . 9 (𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}} ↔ {𝑥𝜑} = {𝑦})
87biimpi 216 . . . . . . . 8 (𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}} → {𝑥𝜑} = {𝑦})
98adantl 481 . . . . . . 7 ((𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}) → {𝑥𝜑} = {𝑦})
109eximi 1835 . . . . . 6 (∃𝑦(𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}) → ∃𝑦{𝑥𝜑} = {𝑦})
113, 10sylbi 217 . . . . 5 (𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} → ∃𝑦{𝑥𝜑} = {𝑦})
1211exlimiv 1930 . . . 4 (∃𝑣 𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} → ∃𝑦{𝑥𝜑} = {𝑦})
132, 12sylbi 217 . . 3 ( {𝑤 ∣ {𝑥𝜑} = {𝑤}} ≠ ∅ → ∃𝑦{𝑥𝜑} = {𝑦})
1413necon1bi 2969 . 2 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → {𝑤 ∣ {𝑥𝜑} = {𝑤}} = ∅)
151, 14eqtrid 2789 1 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  c0 4333  {csn 4626   cuni 4907  cio 6512
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-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-nul 4334  df-sn 4627  df-uni 4908  df-iota 6514
This theorem is referenced by:  iotassuni  6533  iotaex  6534  sn-tz6.12-2  42690
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