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Theorem acexmidlem2 5536
Description: Lemma for acexmid 5538. This builds on acexmidlem1 5535 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 4931 sense because it uses ordered pairs as described in opthreg 4307 rather than df-op 3411).

The set 𝐴 is also found in onsucelsucexmidlem 4281.

(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 2328 . . . . 5 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1779 . . . . 5 (∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitr2i 178 . . . 4 ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
4 acexmidlem.c . . . . . . . . 9 𝐶 = {𝐴, 𝐵}
54eleq2i 2120 . . . . . . . 8 (𝑧𝐶𝑧 ∈ {𝐴, 𝐵})
6 vex 2577 . . . . . . . . 9 𝑧 ∈ V
76elpr 3423 . . . . . . . 8 (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴𝑧 = 𝐵))
85, 7bitri 177 . . . . . . 7 (𝑧𝐶 ↔ (𝑧 = 𝐴𝑧 = 𝐵))
9 onsucelsucexmidlem1 4280 . . . . . . . . . . 11 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10 acexmidlem.a . . . . . . . . . . 11 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
119, 10eleqtrri 2129 . . . . . . . . . 10 ∅ ∈ 𝐴
12 elex2 2587 . . . . . . . . . 10 (∅ ∈ 𝐴 → ∃𝑤 𝑤𝐴)
1311, 12ax-mp 7 . . . . . . . . 9 𝑤 𝑤𝐴
14 eleq2 2117 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
1514exbidv 1722 . . . . . . . . 9 (𝑧 = 𝐴 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐴))
1613, 15mpbiri 161 . . . . . . . 8 (𝑧 = 𝐴 → ∃𝑤 𝑤𝑧)
17 p0ex 3966 . . . . . . . . . . . . 13 {∅} ∈ V
1817prid2 3504 . . . . . . . . . . . 12 {∅} ∈ {∅, {∅}}
19 eqid 2056 . . . . . . . . . . . . 13 {∅} = {∅}
2019orci 660 . . . . . . . . . . . 12 ({∅} = {∅} ∨ 𝜑)
21 eqeq1 2062 . . . . . . . . . . . . . 14 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
2221orbi1d 715 . . . . . . . . . . . . 13 (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
2322elrab 2720 . . . . . . . . . . . 12 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
2418, 20, 23mpbir2an 860 . . . . . . . . . . 11 {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25 acexmidlem.b . . . . . . . . . . 11 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
2624, 25eleqtrri 2129 . . . . . . . . . 10 {∅} ∈ 𝐵
27 elex2 2587 . . . . . . . . . 10 ({∅} ∈ 𝐵 → ∃𝑤 𝑤𝐵)
2826, 27ax-mp 7 . . . . . . . . 9 𝑤 𝑤𝐵
29 eleq2 2117 . . . . . . . . . 10 (𝑧 = 𝐵 → (𝑤𝑧𝑤𝐵))
3029exbidv 1722 . . . . . . . . 9 (𝑧 = 𝐵 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐵))
3128, 30mpbiri 161 . . . . . . . 8 (𝑧 = 𝐵 → ∃𝑤 𝑤𝑧)
3216, 31jaoi 646 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐵) → ∃𝑤 𝑤𝑧)
338, 32sylbi 118 . . . . . 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 119 . . . 4 ((𝑧𝐶 ∧ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
373, 36sylan2br 276 . . 3 ((𝑧𝐶 ∧ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3837ralimiaa 2400 . 2 (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3910, 25, 4acexmidlem1 5535 . 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 101  wo 639  wal 1257   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  ∃!wreu 2325  {crab 2327  c0 3251  {csn 3402  {cpr 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135  df-iota 4894  df-riota 5495
This theorem is referenced by:  acexmidlemv  5537
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