Theorem acexmidlemv 6073

Description: Lemma for acexmid 6074.

This is acexmid 6074 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 4671	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On
2		pp0ex 4321	. . . . 5 ⊢ {∅, {∅}} ∈ V
3	2	rabex 4275	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V
4		prexg 4344	. . . 4 ⊢ (({𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On ∧ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V) → {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V)
5	1, 3, 4	mp2an 430	. . 3 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V
6		raleq 2749	. . . 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 1878	. . 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 2877	. 2 ⊢ ∃𝑦∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)
10		eqeq1 2245	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
11	10	orbi1d 803	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = ∅ ∨ 𝜑) ↔ (𝑡 = ∅ ∨ 𝜑)))
12	11	cbvrabv 2820	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = ∅ ∨ 𝜑)}
13		eqeq1 2245	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
14	13	orbi1d 803	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑠 = {∅} ∨ 𝜑) ↔ (𝑡 = {∅} ∨ 𝜑)))
15	14	cbvrabv 2820	. . . 4 ⊢ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = {∅} ∨ 𝜑)}
16		eqid 2238	. . . 4 ⊢ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}
17	12, 15, 16	acexmidlem2 6072	. . 3 ⊢ (∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
18	17	exlimiv 1651	. 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 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∃!wreu 2530  {crab 2532  Vcvv 2821  ∅c0 3520  {csn 3705  {cpr 3706  Oncon0 4503

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-iota 5332  df-riota 6028

This theorem is referenced by:  acexmid  6074