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Theorem acexmidlem2 5866
Description: Lemma for acexmid 5868. This builds on acexmidlem1 5865 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 5214 sense because it uses ordered pairs as described in opthreg 4552 rather than df-op 3600).

The set 𝐴 is also found in onsucelsucexmidlem 4525.

(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 2460 . . . . 5 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1883 . . . . 5 (∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitr2i 185 . . . 4 ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
4 acexmidlem.c . . . . . . . . 9 𝐶 = {𝐴, 𝐵}
54eleq2i 2244 . . . . . . . 8 (𝑧𝐶𝑧 ∈ {𝐴, 𝐵})
6 vex 2740 . . . . . . . . 9 𝑧 ∈ V
76elpr 3612 . . . . . . . 8 (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴𝑧 = 𝐵))
85, 7bitri 184 . . . . . . 7 (𝑧𝐶 ↔ (𝑧 = 𝐴𝑧 = 𝐵))
9 onsucelsucexmidlem1 4524 . . . . . . . . . . 11 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10 acexmidlem.a . . . . . . . . . . 11 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
119, 10eleqtrri 2253 . . . . . . . . . 10 ∅ ∈ 𝐴
12 elex2 2753 . . . . . . . . . 10 (∅ ∈ 𝐴 → ∃𝑤 𝑤𝐴)
1311, 12ax-mp 5 . . . . . . . . 9 𝑤 𝑤𝐴
14 eleq2 2241 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
1514exbidv 1825 . . . . . . . . 9 (𝑧 = 𝐴 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐴))
1613, 15mpbiri 168 . . . . . . . 8 (𝑧 = 𝐴 → ∃𝑤 𝑤𝑧)
17 p0ex 4185 . . . . . . . . . . . . 13 {∅} ∈ V
1817prid2 3698 . . . . . . . . . . . 12 {∅} ∈ {∅, {∅}}
19 eqid 2177 . . . . . . . . . . . . 13 {∅} = {∅}
2019orci 731 . . . . . . . . . . . 12 ({∅} = {∅} ∨ 𝜑)
21 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
2221orbi1d 791 . . . . . . . . . . . . 13 (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
2322elrab 2893 . . . . . . . . . . . 12 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
2418, 20, 23mpbir2an 942 . . . . . . . . . . 11 {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25 acexmidlem.b . . . . . . . . . . 11 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
2624, 25eleqtrri 2253 . . . . . . . . . 10 {∅} ∈ 𝐵
27 elex2 2753 . . . . . . . . . 10 ({∅} ∈ 𝐵 → ∃𝑤 𝑤𝐵)
2826, 27ax-mp 5 . . . . . . . . 9 𝑤 𝑤𝐵
29 eleq2 2241 . . . . . . . . . 10 (𝑧 = 𝐵 → (𝑤𝑧𝑤𝐵))
3029exbidv 1825 . . . . . . . . 9 (𝑧 = 𝐵 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐵))
3128, 30mpbiri 168 . . . . . . . 8 (𝑧 = 𝐵 → ∃𝑤 𝑤𝑧)
3216, 31jaoi 716 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐵) → ∃𝑤 𝑤𝑧)
338, 32sylbi 121 . . . . . 6 (𝑧𝐶 → ∃𝑤 𝑤𝑧)
34 pm2.27 40 . . . . . 6 (∃𝑤 𝑤𝑧 → ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3533, 34syl 14 . . . . 5 (𝑧𝐶 → ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3635imp 124 . . . 4 ((𝑧𝐶 ∧ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
373, 36sylan2br 288 . . 3 ((𝑧𝐶 ∧ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3837ralimiaa 2539 . 2 (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3910, 25, 4acexmidlem1 5865 . 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 104  wo 708  wal 1351   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  ∃!wreu 2457  {crab 2459  c0 3422  {csn 3591  {cpr 3592
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-iord 4363  df-on 4365  df-suc 4368  df-iota 5174  df-riota 5825
This theorem is referenced by:  acexmidlemv  5867
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