Theorem acexmidlemb 5845

Description: Lemma for acexmid 5852. (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 2237	. . 3 ⊢ (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3		0ex 4116	. . . . 5 ⊢ ∅ ∈ V
4	3	prid1 3689	. . . 4 ⊢ ∅ ∈ {∅, {∅}}
5		eqeq1 2177	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
6	5	orbi1d 786	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
7	6	elrab3 2887	. . . 4 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)))
8	4, 7	ax-mp 5	. . 3 ⊢ (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))
9	2, 8	bitri 183	. 2 ⊢ (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑))
10		noel 3418	. . . 4 ⊢ ¬ ∅ ∈ ∅
11	3	snid 3614	. . . . 5 ⊢ ∅ ∈ {∅}
12		eleq2 2234	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
13	11, 12	mpbiri 167	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
14	10, 13	mto 657	. . 3 ⊢ ¬ ∅ = {∅}
15		orel1 720	. . 3 ⊢ (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑))
16	14, 15	ax-mp 5	. 2 ⊢ ((∅ = {∅} ∨ 𝜑) → 𝜑)
17	9, 16	sylbi 120	1 ⊢ (∅ ∈ 𝐵 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 703   = wceq 1348   ∈ wcel 2141  {crab 2452  ∅c0 3414  {csn 3583  {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-nul 3415  df-sn 3589  df-pr 3590

This theorem is referenced by:  acexmidlem1  5849