Theorem bj-nuliotaALT 33592

Description: Alternate proof of bj-nuliota 33591. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6114). 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 4197	. 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
2		iotassuni 6115	. . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥}
3		eq0 4156	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	3	bicomi 216	. . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅)
5	4	abbii 2907	. . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅}
6	5	unieqi 4680	. . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅}
7		df-sn 4398	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
8	7	eqcomi 2786	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅}
9	8	unieqi 4680	. . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅}
10		0ex 5026	. . . . 5 ⊢ ∅ ∈ V
11	10	unisn 4687	. . . 4 ⊢ ∪ {∅} = ∅
12	6, 9, 11	3eqtri 2805	. . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅
13	2, 12	sseqtri 3855	. 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅
14	1, 13	eqssi 3836	1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3  ∀wal 1599   = wceq 1601  {cab 2762  ∅c0 4140  {csn 4397  ∪ cuni 4671  ℩cio 6097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-sn 4398  df-pr 4400  df-uni 4672  df-iota 6099

This theorem is referenced by: (None)