Theorem acexmidlem1 5817

Description: Lemma for acexmid 5820. List the cases identified in acexmidlemcase 5816 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-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
acexmidlem1	⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑣,𝑢   𝜑,𝑥,𝑦,𝑧,𝑣,𝑢

Proof of Theorem acexmidlem1

Step	Hyp	Ref	Expression
1		acexmidlem.a	. . 3 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
2		acexmidlem.b	. . 3 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
3		acexmidlem.c	. . 3 ⊢ 𝐶 = {𝐴, 𝐵}
4	1, 2, 3	acexmidlemcase 5816	. 2 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})))
5	1, 2, 3	acexmidlema 5812	. . . 4 ⊢ ({∅} ∈ 𝐴 → 𝜑)
6	5	orcd 723	. . 3 ⊢ ({∅} ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
7	1, 2, 3	acexmidlemb 5813	. . . 4 ⊢ (∅ ∈ 𝐵 → 𝜑)
8	7	orcd 723	. . 3 ⊢ (∅ ∈ 𝐵 → (𝜑 ∨ ¬ 𝜑))
9	1, 2, 3	acexmidlemab 5815	. . . 4 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ¬ 𝜑)
10	9	olcd 724	. . 3 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → (𝜑 ∨ ¬ 𝜑))
11	6, 8, 10	3jaoi 1285	. 2 ⊢ (({∅} ∈ 𝐴 ∨ ∅ ∈ 𝐵 ∨ ((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅})) → (𝜑 ∨ ¬ 𝜑))
12	4, 11	syl 14	1 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   ∨ w3o 962   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  ∃!wreu 2437  {crab 2439  ∅c0 3394  {csn 3560  {cpr 3561  ℩crio 5776

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-iota 5134  df-riota 5777

This theorem is referenced by:  acexmidlem2  5818