Theorem bj-nuliotaALT 37638

Description: Alternate proof of bj-nuliota 37637. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6516). 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 6511	. . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥}
3		eq0 4303	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	bicomi 227	. . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
5	4	abbii 2828	. . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅}
6	5	unieqi 4883	. . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅}
7		df-sn 4589	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
8	7	eqcomi 2770	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅}
9	8	unieqi 4883	. . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅}
10		0ex 5269	. . . . 5 ⊢ ∅ ∈ V
11	10	unisn 4890	. . . 4 ⊢ ∪ {∅} = ∅
12	6, 9, 11	3eqtri 2788	. . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅
13	2, 12	sseqtri 3984	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅
14	1, 13	eqssi 3952	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3  ∀wal 1566   = wceq 1568  {cab 2739  ∅c0 4285  {csn 4588  ∪ cuni 4871  ℩cio 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4589  df-pr 4591  df-uni 4872  df-iota 6492

This theorem is referenced by: (None)