Theorem acexmidlema 5879

Description: Lemma for acexmid 5887. (Contributed by Jim Kingdon, 6-Aug-2019.)

Hypotheses

Ref	Expression
acexmidlem.a	⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
acexmidlem.b	⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
acexmidlem.c	⊢ 𝐶 = {𝐴, 𝐵}

Assertion

Ref	Expression
acexmidlema	⊢ ({∅} ∈ 𝐴 → 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Proof of Theorem acexmidlema

Step	Hyp	Ref	Expression
1		acexmidlem.a	. . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
2	1	eleq2i 2254	. . 3 ⊢ ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3		p0ex 4200	. . . . 5 ⊢ {∅} ∈ V
4	3	prid2 3711	. . . 4 ⊢ {∅} ∈ {∅, {∅}}
5		eqeq1 2194	. . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅))
6	5	orbi1d 792	. . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑)))
7	6	elrab3 2906	. . . 4 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)))
8	4, 7	ax-mp 5	. . 3 ⊢ ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))
9	2, 8	bitri 184	. 2 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} = ∅ ∨ 𝜑))
10		noel 3438	. . . 4 ⊢ ¬ ∅ ∈ ∅
11		0ex 4142	. . . . . 6 ⊢ ∅ ∈ V
12	11	snid 3635	. . . . 5 ⊢ ∅ ∈ {∅}
13		eleq2 2251	. . . . 5 ⊢ ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅))
14	12, 13	mpbii 148	. . . 4 ⊢ ({∅} = ∅ → ∅ ∈ ∅)
15	10, 14	mto 663	. . 3 ⊢ ¬ {∅} = ∅
16		orel1 726	. . 3 ⊢ (¬ {∅} = ∅ → (({∅} = ∅ ∨ 𝜑) → 𝜑))
17	15, 16	ax-mp 5	. 2 ⊢ (({∅} = ∅ ∨ 𝜑) → 𝜑)
18	9, 17	sylbi 121	1 ⊢ ({∅} ∈ 𝐴 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709   = wceq 1363   ∈ wcel 2158  {crab 2469  ∅c0 3434  {csn 3604  {cpr 3605

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611

This theorem is referenced by:  acexmidlem1  5884