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Theorem bj-nuliotaALT 32694
 Description: Alternate proof of bj-nuliota 32693. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 5830). 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 3949 . 2 ∅ ⊆ (℩𝑥𝑦 ¬ 𝑦𝑥)
2 iotassuni 5831 . . 3 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥}
3 eq0 3910 . . . . . . 7 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43bicomi 214 . . . . . 6 (∀𝑦 ¬ 𝑦𝑥𝑥 = ∅)
54abbii 2736 . . . . 5 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
65unieqi 4416 . . . 4 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
7 df-sn 4154 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
87eqcomi 2630 . . . . 5 {𝑥𝑥 = ∅} = {∅}
98unieqi 4416 . . . 4 {𝑥𝑥 = ∅} = {∅}
10 0ex 4755 . . . . 5 ∅ ∈ V
1110unisn 4422 . . . 4 {∅} = ∅
126, 9, 113eqtri 2647 . . 3 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = ∅
132, 12sseqtri 3621 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ ∅
141, 13eqssi 3603 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1478   = wceq 1480  {cab 2607  ∅c0 3896  {csn 4153  ∪ cuni 4407  ℩cio 5813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5815 This theorem is referenced by: (None)
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