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Theorem iotanul2 6533
Description: Version of iotanul 6541 using df-iota 6516 instead of dfiota2 6517. (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 6516 . 2 (℩𝑥𝜑) = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
2 n0 4359 . . . 4 ( {𝑤 ∣ {𝑥𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}})
3 eluni 4915 . . . . . 6 (𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} ↔ ∃𝑦(𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}))
4 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
5 sneq 4641 . . . . . . . . . . 11 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2746 . . . . . . . . . 10 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6elab 3681 . . . . . . . . 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 1832 . . . . . 6 (∃𝑦(𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}) → ∃𝑦{𝑥𝜑} = {𝑦})
113, 10sylbi 217 . . . . 5 (𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} → ∃𝑦{𝑥𝜑} = {𝑦})
1211exlimiv 1928 . . . 4 (∃𝑣 𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} → ∃𝑦{𝑥𝜑} = {𝑦})
132, 12sylbi 217 . . 3 ( {𝑤 ∣ {𝑥𝜑} = {𝑤}} ≠ ∅ → ∃𝑦{𝑥𝜑} = {𝑦})
1413necon1bi 2967 . 2 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → {𝑤 ∣ {𝑥𝜑} = {𝑤}} = ∅)
151, 14eqtrid 2787 1 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  c0 4339  {csn 4631   cuni 4912  cio 6514
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-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-nul 4340  df-sn 4632  df-uni 4913  df-iota 6516
This theorem is referenced by:  iotassuni  6535  iotaex  6536  sn-tz6.12-2  42667
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