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Theorem acexmidlem2 5588
Description: Lemma for acexmid 5590. This builds on acexmidlem1 5587 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 4971 sense because it uses ordered pairs as described in opthreg 4335 rather than df-op 3431).

The set 𝐴 is also found in onsucelsucexmidlem 4308.

(Contributed by Jim Kingdon, 5-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
acexmidlem2 (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢   𝜑,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem acexmidlem2
StepHypRef Expression
1 df-ral 2358 . . . . 5 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1806 . . . . 5 (∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitr2i 183 . . . 4 ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
4 acexmidlem.c . . . . . . . . 9 𝐶 = {𝐴, 𝐵}
54eleq2i 2149 . . . . . . . 8 (𝑧𝐶𝑧 ∈ {𝐴, 𝐵})
6 vex 2615 . . . . . . . . 9 𝑧 ∈ V
76elpr 3443 . . . . . . . 8 (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴𝑧 = 𝐵))
85, 7bitri 182 . . . . . . 7 (𝑧𝐶 ↔ (𝑧 = 𝐴𝑧 = 𝐵))
9 onsucelsucexmidlem1 4307 . . . . . . . . . . 11 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10 acexmidlem.a . . . . . . . . . . 11 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
119, 10eleqtrri 2158 . . . . . . . . . 10 ∅ ∈ 𝐴
12 elex2 2626 . . . . . . . . . 10 (∅ ∈ 𝐴 → ∃𝑤 𝑤𝐴)
1311, 12ax-mp 7 . . . . . . . . 9 𝑤 𝑤𝐴
14 eleq2 2146 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
1514exbidv 1748 . . . . . . . . 9 (𝑧 = 𝐴 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐴))
1613, 15mpbiri 166 . . . . . . . 8 (𝑧 = 𝐴 → ∃𝑤 𝑤𝑧)
17 p0ex 3987 . . . . . . . . . . . . 13 {∅} ∈ V
1817prid2 3523 . . . . . . . . . . . 12 {∅} ∈ {∅, {∅}}
19 eqid 2083 . . . . . . . . . . . . 13 {∅} = {∅}
2019orci 683 . . . . . . . . . . . 12 ({∅} = {∅} ∨ 𝜑)
21 eqeq1 2089 . . . . . . . . . . . . . 14 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
2221orbi1d 738 . . . . . . . . . . . . 13 (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
2322elrab 2759 . . . . . . . . . . . 12 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
2418, 20, 23mpbir2an 884 . . . . . . . . . . 11 {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25 acexmidlem.b . . . . . . . . . . 11 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
2624, 25eleqtrri 2158 . . . . . . . . . 10 {∅} ∈ 𝐵
27 elex2 2626 . . . . . . . . . 10 ({∅} ∈ 𝐵 → ∃𝑤 𝑤𝐵)
2826, 27ax-mp 7 . . . . . . . . 9 𝑤 𝑤𝐵
29 eleq2 2146 . . . . . . . . . 10 (𝑧 = 𝐵 → (𝑤𝑧𝑤𝐵))
3029exbidv 1748 . . . . . . . . 9 (𝑧 = 𝐵 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐵))
3128, 30mpbiri 166 . . . . . . . 8 (𝑧 = 𝐵 → ∃𝑤 𝑤𝑧)
3216, 31jaoi 669 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐵) → ∃𝑤 𝑤𝑧)
338, 32sylbi 119 . . . . . 6 (𝑧𝐶 → ∃𝑤 𝑤𝑧)
34 pm2.27 39 . . . . . 6 (∃𝑤 𝑤𝑧 → ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3533, 34syl 14 . . . . 5 (𝑧𝐶 → ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3635imp 122 . . . 4 ((𝑧𝐶 ∧ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
373, 36sylan2br 282 . . 3 ((𝑧𝐶 ∧ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3837ralimiaa 2431 . 2 (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3910, 25, 4acexmidlem1 5587 . 2 (∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
4038, 39syl 14 1 (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  wal 1283   = wceq 1285  wex 1422  wcel 1434  wral 2353  wrex 2354  ∃!wreu 2355  {crab 2357  c0 3269  {csn 3422  {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-iord 4157  df-on 4159  df-suc 4162  df-iota 4934  df-riota 5547
This theorem is referenced by:  acexmidlemv  5589
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