Theorem acexmidlemv 5998

Description: Lemma for acexmid 5999.

This is acexmid 5999 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 4620	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On
2		pp0ex 4272	. . . . 5 ⊢ {∅, {∅}} ∈ V
3	2	rabex 4227	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V
4		prexg 4294	. . . 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 2728	. . . 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 1871	. . 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 2855	. 2 ⊢ ∃𝑦∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)
10		eqeq1 2236	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
11	10	orbi1d 796	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = ∅ ∨ 𝜑) ↔ (𝑡 = ∅ ∨ 𝜑)))
12	11	cbvrabv 2798	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = ∅ ∨ 𝜑)}
13		eqeq1 2236	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
14	13	orbi1d 796	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = {∅} ∨ 𝜑) ↔ (𝑡 = {∅} ∨ 𝜑)))
15	14	cbvrabv 2798	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = {∅} ∨ 𝜑)}
16		eqid 2229	. . . 4 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}
17	12, 15, 16	acexmidlem2 5997	. . 3 ⊢ (∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
18	17	exlimiv 1644	. 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 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510  {crab 2512  Vcvv 2799  ∅c0 3491  {csn 3666  {cpr 3667  Oncon0 4453

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-iota 5277  df-riota 5953

This theorem is referenced by:  acexmid  5999