Theorem bj-nuliotaALT 37272

Description: Alternate proof of bj-nuliota 37271. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6473). 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	⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nuliotaALT

Step	Hyp	Ref	Expression
1		0ss 4353	. 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
2		iotassuni 6468	. . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥}
3		eq0 4303	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	bicomi 224	. . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
5	4	abbii 2804	. . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅}
6	5	unieqi 4876	. . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅}
7		df-sn 4582	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
8	7	eqcomi 2746	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅}
9	8	unieqi 4876	. . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅}
10		0ex 5253	. . . . 5 ⊢ ∅ ∈ V
11	10	unisn 4883	. . . 4 ⊢ ∪ {∅} = ∅
12	6, 9, 11	3eqtri 2764	. . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅
13	2, 12	sseqtri 3983	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅
14	1, 13	eqssi 3951	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3  ∀wal 1540   = wceq 1542  {cab 2715  ∅c0 4286  {csn 4581  ∪ cuni 4864  ℩cio 6447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6449

This theorem is referenced by: (None)