Theorem acexmidlemv 6026

Description: Lemma for acexmid 6027.

This is acexmid 6027 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 4633	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On
2		pp0ex 4285	. . . . 5 ⊢ {∅, {∅}} ∈ V
3	2	rabex 4239	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V
4		prexg 4307	. . . 4 ⊢ (({𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On ∧ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V) → {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V)
5	1, 3, 4	mp2an 426	. . 3 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V
6		raleq 2731	. . . 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 1873	. . 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 2859	. 2 ⊢ ∃𝑦∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)
10		eqeq1 2238	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
11	10	orbi1d 799	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = ∅ ∨ 𝜑) ↔ (𝑡 = ∅ ∨ 𝜑)))
12	11	cbvrabv 2802	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = ∅ ∨ 𝜑)}
13		eqeq1 2238	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
14	13	orbi1d 799	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = {∅} ∨ 𝜑) ↔ (𝑡 = {∅} ∨ 𝜑)))
15	14	cbvrabv 2802	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = {∅} ∨ 𝜑)}
16		eqid 2231	. . . 4 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}
17	12, 15, 16	acexmidlem2 6025	. . 3 ⊢ (∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
18	17	exlimiv 1647	. 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 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  ∃!wreu 2513  {crab 2515  Vcvv 2803  ∅c0 3496  {csn 3673  {cpr 3674  Oncon0 4466

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474  df-iota 5293  df-riota 5981

This theorem is referenced by:  acexmid  6027