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Theorem bj-nuliotaALT 32211
 Description: Alternate proof of bj-nuliota 32210. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 5783). 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 3924 . 2 ∅ ⊆ (℩𝑥𝑦 ¬ 𝑦𝑥)
2 iotassuni 5784 . . 3 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥}
3 eq0 3888 . . . . . . 7 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
43bicomi 213 . . . . . 6 (∀𝑦 ¬ 𝑦𝑥𝑥 = ∅)
54abbii 2726 . . . . 5 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
65unieqi 4381 . . . 4 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = {𝑥𝑥 = ∅}
7 df-sn 4126 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
87eqcomi 2619 . . . . 5 {𝑥𝑥 = ∅} = {∅}
98unieqi 4381 . . . 4 {𝑥𝑥 = ∅} = {∅}
10 0ex 4718 . . . . 5 ∅ ∈ V
1110unisn 4387 . . . 4 {∅} = ∅
126, 9, 113eqtri 2636 . . 3 {𝑥 ∣ ∀𝑦 ¬ 𝑦𝑥} = ∅
132, 12sseqtri 3600 . 2 (℩𝑥𝑦 ¬ 𝑦𝑥) ⊆ ∅
141, 13eqssi 3584 1 ∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1473   = wceq 1475  {cab 2596  ∅c0 3874  {csn 4125  ∪ cuni 4372  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768 This theorem is referenced by: (None)
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