Theorem acexmidlemb 5759

Description: Lemma for acexmid 5766. (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 2204	. . 3 ⊢ (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3		0ex 4050	. . . . 5 ⊢ ∅ ∈ V
4	3	prid1 3624	. . . 4 ⊢ ∅ ∈ {∅, {∅}}
5		eqeq1 2144	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
6	5	orbi1d 780	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
7	6	elrab3 2836	. . . 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 3362	. . . 4 ⊢ ¬ ∅ ∈ ∅
11	3	snid 3551	. . . . 5 ⊢ ∅ ∈ {∅}
12		eleq2 2201	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
13	11, 12	mpbiri 167	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
14	10, 13	mto 651	. . 3 ⊢ ¬ ∅ = {∅}
15		orel1 714	. . 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 697   = wceq 1331   ∈ wcel 1480  {crab 2418  ∅c0 3358  {csn 3522  {cpr 3523

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-nul 3359  df-sn 3528  df-pr 3529

This theorem is referenced by:  acexmidlem1  5763