Theorem acexmidlemb 5883

Description: Lemma for acexmid 5890. (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
acexmidlemb	⊢ (∅ ∈ 𝐵 → 𝜑)

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

Proof of Theorem acexmidlemb

Step	Hyp	Ref	Expression
1		acexmidlem.b	. . . 4 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
2	1	eleq2i 2256	. . 3 ⊢ (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3		0ex 4145	. . . . 5 ⊢ ∅ ∈ V
4	3	prid1 3713	. . . 4 ⊢ ∅ ∈ {∅, {∅}}
5		eqeq1 2196	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
6	5	orbi1d 792	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
7	6	elrab3 2909	. . . 4 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)))
8	4, 7	ax-mp 5	. . 3 ⊢ (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))
9	2, 8	bitri 184	. 2 ⊢ (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑))
10		noel 3441	. . . 4 ⊢ ¬ ∅ ∈ ∅
11	3	snid 3638	. . . . 5 ⊢ ∅ ∈ {∅}
12		eleq2 2253	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
13	11, 12	mpbiri 168	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
14	10, 13	mto 663	. . 3 ⊢ ¬ ∅ = {∅}
15		orel1 726	. . 3 ⊢ (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑))
16	14, 15	ax-mp 5	. 2 ⊢ ((∅ = {∅} ∨ 𝜑) → 𝜑)
17	9, 16	sylbi 121	1 ⊢ (∅ ∈ 𝐵 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2160  {crab 2472  ∅c0 3437  {csn 3607  {cpr 3608

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-pr 3614

This theorem is referenced by:  acexmidlem1  5887