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Theorem bj-nuliotaALT 34245
Description: Alternate proof of bj-nuliota 34244. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 6326). 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 4347 . 2 ∅ ⊆ (℩𝑥𝑦 ¬ 𝑦𝑥)
2 iotassuni 6327 . . 3 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥}
3 eq0 4305 . . . . . . 7 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43bicomi 225 . . . . . 6 (∀𝑦 ¬ 𝑦𝑥𝑥 = ∅)
54abbii 2883 . . . . 5 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
65unieqi 4839 . . . 4 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
7 df-sn 4558 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
87eqcomi 2827 . . . . 5 {𝑥𝑥 = ∅} = {∅}
98unieqi 4839 . . . 4 {𝑥𝑥 = ∅} = {∅}
10 0ex 5202 . . . . 5 ∅ ∈ V
1110unisn 4846 . . . 4 {∅} = ∅
126, 9, 113eqtri 2845 . . 3 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = ∅
132, 12sseqtri 4000 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ ∅
141, 13eqssi 3980 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1526   = wceq 1528  {cab 2796  c0 4288  {csn 4557   cuni 4830  cio 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307
This theorem is referenced by: (None)
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