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Theorem sn-iotanul 40181
Description: Version of iotanul 6406 using df-iota 6386 instead of dfiota2 6387. (Contributed by SN, 6-Nov-2024.)
Assertion
Ref Expression
sn-iotanul (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotanul
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iota 6386 . 2 (℩𝑥𝜑) = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
2 n0 4282 . . . 4 ( {𝑤 ∣ {𝑥𝜑} = {𝑤}} ≠ ∅ ↔ ∃𝑣 𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}})
3 eluni 4844 . . . . . 6 (𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} ↔ ∃𝑦(𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}))
4 vex 3435 . . . . . . . . . 10 𝑦 ∈ V
5 sneq 4573 . . . . . . . . . . 11 (𝑤 = 𝑦 → {𝑤} = {𝑦})
65eqeq2d 2749 . . . . . . . . . 10 (𝑤 = 𝑦 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑦}))
74, 6elab 3610 . . . . . . . . 9 (𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}} ↔ {𝑥𝜑} = {𝑦})
87biimpi 215 . . . . . . . 8 (𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}} → {𝑥𝜑} = {𝑦})
98adantl 482 . . . . . . 7 ((𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}) → {𝑥𝜑} = {𝑦})
109eximi 1837 . . . . . 6 (∃𝑦(𝑣𝑦𝑦 ∈ {𝑤 ∣ {𝑥𝜑} = {𝑤}}) → ∃𝑦{𝑥𝜑} = {𝑦})
113, 10sylbi 216 . . . . 5 (𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} → ∃𝑦{𝑥𝜑} = {𝑦})
1211exlimiv 1933 . . . 4 (∃𝑣 𝑣 {𝑤 ∣ {𝑥𝜑} = {𝑤}} → ∃𝑦{𝑥𝜑} = {𝑦})
132, 12sylbi 216 . . 3 ( {𝑤 ∣ {𝑥𝜑} = {𝑤}} ≠ ∅ → ∃𝑦{𝑥𝜑} = {𝑦})
1413necon1bi 2972 . 2 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → {𝑤 ∣ {𝑥𝜑} = {𝑤}} = ∅)
151, 14eqtrid 2790 1 (¬ ∃𝑦{𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wne 2943  c0 4258  {csn 4563   cuni 4841  cio 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3433  df-dif 3891  df-nul 4259  df-sn 4564  df-uni 4842  df-iota 6386
This theorem is referenced by:  sn-iotassuni  40183  sn-iotaex  40184
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