Theorem bj-nuliotaALT 34345

Description: Alternate proof of bj-nuliota 34344. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6327). 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 4349	. 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
2		iotassuni 6328	. . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥}
3		eq0 4307	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	bicomi 226	. . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
5	4	abbii 2886	. . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅}
6	5	unieqi 4840	. . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅}
7		df-sn 4561	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
8	7	eqcomi 2830	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅}
9	8	unieqi 4840	. . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅}
10		0ex 5203	. . . . 5 ⊢ ∅ ∈ V
11	10	unisn 4847	. . . 4 ⊢ ∪ {∅} = ∅
12	6, 9, 11	3eqtri 2848	. . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅
13	2, 12	sseqtri 4002	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅
14	1, 13	eqssi 3982	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3  ∀wal 1531   = wceq 1533  {cab 2799  ∅c0 4290  {csn 4560  ∪ cuni 4831  ℩cio 6306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4561  df-pr 4563  df-uni 4832  df-iota 6308

This theorem is referenced by: (None)