Theorem acexmidlemb 5504

Description: Lemma for acexmid 5511. (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 2104	. . 3 ⊢ (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
3		0ex 3884	. . . . 5 ⊢ ∅ ∈ V
4	3	prid1 3476	. . . 4 ⊢ ∅ ∈ {∅, {∅}}
5		eqeq1 2046	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅}))
6	5	orbi1d 705	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑)))
7	6	elrab3 2699	. . . 4 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)))
8	4, 7	ax-mp 7	. . 3 ⊢ (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))
9	2, 8	bitri 173	. 2 ⊢ (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑))
10		noel 3228	. . . 4 ⊢ ¬ ∅ ∈ ∅
11	3	snid 3402	. . . . 5 ⊢ ∅ ∈ {∅}
12		eleq2 2101	. . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅}))
13	11, 12	mpbiri 157	. . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅)
14	10, 13	mto 588	. . 3 ⊢ ¬ ∅ = {∅}
15		orel1 644	. . 3 ⊢ (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑))
16	14, 15	ax-mp 7	. 2 ⊢ ((∅ = {∅} ∨ 𝜑) → 𝜑)
17	9, 16	sylbi 114	1 ⊢ (∅ ∈ 𝐵 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  {crab 2310  ∅c0 3224  {csn 3375  {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382

This theorem is referenced by:  acexmidlem1  5508