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Theorem bj-nuliotaALT 37059
Description: Alternate proof of bj-nuliota 37058. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6539). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-nuliotaALT ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nuliotaALT
StepHypRef Expression
1 0ss 4400 . 2 ∅ ⊆ (℩𝑥𝑦 ¬ 𝑦𝑥)
2 iotassuni 6533 . . 3 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥}
3 eq0 4350 . . . . . . 7 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43bicomi 224 . . . . . 6 (∀𝑦 ¬ 𝑦𝑥𝑥 = ∅)
54abbii 2809 . . . . 5 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
65unieqi 4919 . . . 4 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
7 df-sn 4627 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
87eqcomi 2746 . . . . 5 {𝑥𝑥 = ∅} = {∅}
98unieqi 4919 . . . 4 {𝑥𝑥 = ∅} = {∅}
10 0ex 5307 . . . . 5 ∅ ∈ V
1110unisn 4926 . . . 4 {∅} = ∅
126, 9, 113eqtri 2769 . . 3 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = ∅
132, 12sseqtri 4032 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ ∅
141, 13eqssi 4000 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1538   = wceq 1540  {cab 2714  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  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514
This theorem is referenced by: (None)
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