Theorem acexmidlemv 5840

Description: Lemma for acexmid 5841.

This is acexmid 5841 with additional disjoint variable conditions, most notably between 𝜑 and 𝑥.

(Contributed by Jim Kingdon, 6-Aug-2019.)

Hypothesis

Ref	Expression
acexmidlemv.choice	⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)

Assertion

Ref	Expression
acexmidlemv	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝜑,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem acexmidlemv

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucelsucexmidlem 4506	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On
2		pp0ex 4168	. . . . 5 ⊢ {∅, {∅}} ∈ V
3	2	rabex 4126	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V
4		prexg 4189	. . . 4 ⊢ (({𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On ∧ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V) → {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V)
5	1, 3, 4	mp2an 423	. . 3 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V
6		raleq 2661	. . . 4 ⊢ (𝑥 = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
7	6	exbidv 1813	. . 3 ⊢ (𝑥 = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} → (∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∃𝑦∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
8		acexmidlemv.choice	. . 3 ⊢ ∃𝑦∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)
9	5, 7, 8	vtocl 2780	. 2 ⊢ ∃𝑦∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)
10		eqeq1 2172	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
11	10	orbi1d 781	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = ∅ ∨ 𝜑) ↔ (𝑡 = ∅ ∨ 𝜑)))
12	11	cbvrabv 2725	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = ∅ ∨ 𝜑)}
13		eqeq1 2172	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
14	13	orbi1d 781	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = {∅} ∨ 𝜑) ↔ (𝑡 = {∅} ∨ 𝜑)))
15	14	cbvrabv 2725	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = {∅} ∨ 𝜑)}
16		eqid 2165	. . . 4 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}
17	12, 15, 16	acexmidlem2 5839	. . 3 ⊢ (∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
18	17	exlimiv 1586	. 2 ⊢ (∃𝑦∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
19	9, 18	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 698   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  ∃!wreu 2446  {crab 2448  Vcvv 2726  ∅c0 3409  {csn 3576  {cpr 3577  Oncon0 4341

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349  df-iota 5153  df-riota 5798

This theorem is referenced by:  acexmid  5841