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Theorem bj-nuliotaALT 37041
Description: Alternate proof of bj-nuliota 37040. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6541). 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 4406 . 2 ∅ ⊆ (℩𝑥𝑦 ¬ 𝑦𝑥)
2 iotassuni 6535 . . 3 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥}
3 eq0 4356 . . . . . . 7 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43bicomi 224 . . . . . 6 (∀𝑦 ¬ 𝑦𝑥𝑥 = ∅)
54abbii 2807 . . . . 5 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
65unieqi 4924 . . . 4 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
7 df-sn 4632 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
87eqcomi 2744 . . . . 5 {𝑥𝑥 = ∅} = {∅}
98unieqi 4924 . . . 4 {𝑥𝑥 = ∅} = {∅}
10 0ex 5313 . . . . 5 ∅ ∈ V
1110unisn 4931 . . . 4 {∅} = ∅
126, 9, 113eqtri 2767 . . 3 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = ∅
132, 12sseqtri 4032 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ ∅
141, 13eqssi 4012 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1535   = wceq 1537  {cab 2712  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  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by: (None)
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