Theorem bj-nuliotaALT 37548

Description: Alternate proof of bj-nuliota 37547. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6503). 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 4356	. 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
2		iotassuni 6498	. . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥}
3		eq0 4304	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	bicomi 226	. . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
5	4	abbii 2831	. . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅}
6	5	unieqi 4879	. . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅}
7		df-sn 4585	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
8	7	eqcomi 2773	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅}
9	8	unieqi 4879	. . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅}
10		0ex 5259	. . . . 5 ⊢ ∅ ∈ V
11	10	unisn 4886	. . . 4 ⊢ ∪ {∅} = ∅
12	6, 9, 11	3eqtri 2791	. . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅
13	2, 12	sseqtri 3986	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅
14	1, 13	eqssi 3954	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3  ∀wal 1560   = wceq 1562  {cab 2742  ∅c0 4287  {csn 4584  ∪ cuni 4867  ℩cio 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6479

This theorem is referenced by: (None)