Theorem acexmidlem1 5686

Description: Lemma for acexmid 5689. List the cases identified in acexmidlemcase 5685 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 5685	. 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 5681	. . . 4 ⊢ ({∅} ∈ 𝐴 → 𝜑)
6	5	orcd 690	. . 3 ⊢ ({∅} ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
7	1, 2, 3	acexmidlemb 5682	. . . 4 ⊢ (∅ ∈ 𝐵 → 𝜑)
8	7	orcd 690	. . 3 ⊢ (∅ ∈ 𝐵 → (𝜑 ∨ ¬ 𝜑))
9	1, 2, 3	acexmidlemab 5684	. . . 4 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → ¬ 𝜑)
10	9	olcd 691	. . 3 ⊢ (((℩𝑣 ∈ 𝐴 ∃𝑢 ∈ 𝑦 (𝐴 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = ∅ ∧ (℩𝑣 ∈ 𝐵 ∃𝑢 ∈ 𝑦 (𝐵 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) = {∅}) → (𝜑 ∨ ¬ 𝜑))
11	6, 8, 10	3jaoi 1246	. 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 667   ∨ w3o 926   = wceq 1296   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  ∃!wreu 2372  {crab 2374  ∅c0 3302  {csn 3466  {cpr 3467  ℩crio 5645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222  df-iota 5014  df-riota 5646

This theorem is referenced by:  acexmidlem2  5687