Theorem onsucelsucexmidlem1 4620

Description: Lemma for onsucelsucexmid 4622. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
onsucelsucexmidlem1	⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}

Distinct variable group:   𝜑,𝑥

Proof of Theorem onsucelsucexmidlem1

Step	Hyp	Ref	Expression
1		0ex 4211	. . 3 ⊢ ∅ ∈ V
2	1	prid1 3772	. 2 ⊢ ∅ ∈ {∅, {∅}}
3		eqid 2229	. . 3 ⊢ ∅ = ∅
4	3	orci 736	. 2 ⊢ (∅ = ∅ ∨ 𝜑)
5		eqeq1 2236	. . . 4 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
6	5	orbi1d 796	. . 3 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ 𝜑) ↔ (∅ = ∅ ∨ 𝜑)))
7	6	elrab 2959	. 2 ⊢ (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = ∅ ∨ 𝜑)))
8	2, 4, 7	mpbir2an 948	1 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}

Syntax hints:   ∨ wo 713   = wceq 1395   ∈ wcel 2200  {crab 2512  ∅c0 3491  {csn 3666  {cpr 3667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4210

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-pr 3673

This theorem is referenced by:  onsucelsucexmidlem  4621  onsucelsucexmid  4622  acexmidlem2  5998