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Theorem acexmidlem2 5737
Description: Lemma for acexmid 5739. This builds on acexmidlem1 5736 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 5093 sense because it uses ordered pairs as described in opthreg 4439 rather than df-op 3504).

The set 𝐴 is also found in onsucelsucexmidlem 4412.

(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 2396 . . . . 5 (∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2 19.23v 1837 . . . . 5 (∀𝑤(𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
31, 2bitr2i 184 . . . 4 ((∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
4 acexmidlem.c . . . . . . . . 9 𝐶 = {𝐴, 𝐵}
54eleq2i 2182 . . . . . . . 8 (𝑧𝐶𝑧 ∈ {𝐴, 𝐵})
6 vex 2661 . . . . . . . . 9 𝑧 ∈ V
76elpr 3516 . . . . . . . 8 (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴𝑧 = 𝐵))
85, 7bitri 183 . . . . . . 7 (𝑧𝐶 ↔ (𝑧 = 𝐴𝑧 = 𝐵))
9 onsucelsucexmidlem1 4411 . . . . . . . . . . 11 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10 acexmidlem.a . . . . . . . . . . 11 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
119, 10eleqtrri 2191 . . . . . . . . . 10 ∅ ∈ 𝐴
12 elex2 2674 . . . . . . . . . 10 (∅ ∈ 𝐴 → ∃𝑤 𝑤𝐴)
1311, 12ax-mp 5 . . . . . . . . 9 𝑤 𝑤𝐴
14 eleq2 2179 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑤𝑧𝑤𝐴))
1514exbidv 1779 . . . . . . . . 9 (𝑧 = 𝐴 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐴))
1613, 15mpbiri 167 . . . . . . . 8 (𝑧 = 𝐴 → ∃𝑤 𝑤𝑧)
17 p0ex 4080 . . . . . . . . . . . . 13 {∅} ∈ V
1817prid2 3598 . . . . . . . . . . . 12 {∅} ∈ {∅, {∅}}
19 eqid 2115 . . . . . . . . . . . . 13 {∅} = {∅}
2019orci 703 . . . . . . . . . . . 12 ({∅} = {∅} ∨ 𝜑)
21 eqeq1 2122 . . . . . . . . . . . . . 14 (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
2221orbi1d 763 . . . . . . . . . . . . 13 (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
2322elrab 2811 . . . . . . . . . . . 12 ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
2418, 20, 23mpbir2an 909 . . . . . . . . . . 11 {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25 acexmidlem.b . . . . . . . . . . 11 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
2624, 25eleqtrri 2191 . . . . . . . . . 10 {∅} ∈ 𝐵
27 elex2 2674 . . . . . . . . . 10 ({∅} ∈ 𝐵 → ∃𝑤 𝑤𝐵)
2826, 27ax-mp 5 . . . . . . . . 9 𝑤 𝑤𝐵
29 eleq2 2179 . . . . . . . . . 10 (𝑧 = 𝐵 → (𝑤𝑧𝑤𝐵))
3029exbidv 1779 . . . . . . . . 9 (𝑧 = 𝐵 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝐵))
3128, 30mpbiri 167 . . . . . . . 8 (𝑧 = 𝐵 → ∃𝑤 𝑤𝑧)
3216, 31jaoi 688 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐵) → ∃𝑤 𝑤𝑧)
338, 32sylbi 120 . . . . . 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 123 . . . 4 ((𝑧𝐶 ∧ (∃𝑤 𝑤𝑧 → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
373, 36sylan2br 284 . . 3 ((𝑧𝐶 ∧ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)) → ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3837ralimiaa 2469 . 2 (∀𝑧𝐶𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → ∀𝑧𝐶 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
3910, 25, 4acexmidlem1 5736 . 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 103  wo 680  wal 1312   = wceq 1314  wex 1451  wcel 1463  wral 2391  wrex 2392  ∃!wreu 2393  {crab 2395  c0 3331  {csn 3495  {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261  df-iota 5056  df-riota 5696
This theorem is referenced by:  acexmidlemv  5738
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