Theorem acexmidlem2 6055

Description: Lemma for acexmid 6057. This builds on acexmidlem1 6054 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 5359 sense because it uses ordered pairs as described in opthreg 4683 rather than df-op 3703).

The set 𝐴 is also found in onsucelsucexmidlem 4656.

(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	⊢ (∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (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

Step	Hyp	Ref	Expression
1		df-ral 2527	. . . . 5 ⊢ (∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) ↔ ∀𝑤(𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
2		19.23v 1932	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ↔ (∃𝑤 𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
3	1, 2	bitr2i 185	. . . 4 ⊢ ((∃𝑤 𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) ↔ ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
4		acexmidlem.c	. . . . . . . . 9 ⊢ 𝐶 = {𝐴, 𝐵}
5	4	eleq2i 2301	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ {𝐴, 𝐵})
6		vex 2818	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7	6	elpr 3715	. . . . . . . 8 ⊢ (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
8	5, 7	bitri 184	. . . . . . 7 ⊢ (𝑧 ∈ 𝐶 ↔ (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
9		onsucelsucexmidlem1 4655	. . . . . . . . . . 11 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10		acexmidlem.a	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
11	9, 10	eleqtrri 2310	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
12		elex2 2832	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ 𝐴
14		eleq2 2298	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐴))
15	14	exbidv 1874	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝐴))
16	13, 15	mpbiri 168	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ∃𝑤 𝑤 ∈ 𝑧)
17		p0ex 4306	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
18	17	prid2 3803	. . . . . . . . . . . 12 ⊢ {∅} ∈ {∅, {∅}}
19		eqid 2234	. . . . . . . . . . . . 13 ⊢ {∅} = {∅}
20	19	orci 739	. . . . . . . . . . . 12 ⊢ ({∅} = {∅} ∨ 𝜑)
21		eqeq1 2241	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
22	21	orbi1d 799	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
23	22	elrab 2976	. . . . . . . . . . . 12 ⊢ ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
24	18, 20, 23	mpbir2an 951	. . . . . . . . . . 11 ⊢ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25		acexmidlem.b	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
26	24, 25	eleqtrri 2310	. . . . . . . . . 10 ⊢ {∅} ∈ 𝐵
27		elex2 2832	. . . . . . . . . 10 ⊢ ({∅} ∈ 𝐵 → ∃𝑤 𝑤 ∈ 𝐵)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ 𝐵
29		eleq2 2298	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵))
30	29	exbidv 1874	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝐵))
31	28, 30	mpbiri 168	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ∃𝑤 𝑤 ∈ 𝑧)
32	16, 31	jaoi 724	. . . . . . 7 ⊢ ((𝑧 = 𝐴 ∨ 𝑧 = 𝐵) → ∃𝑤 𝑤 ∈ 𝑧)
33	8, 32	sylbi 121	. . . . . 6 ⊢ (𝑧 ∈ 𝐶 → ∃𝑤 𝑤 ∈ 𝑧)
34		pm2.27 40	. . . . . 6 ⊢ (∃𝑤 𝑤 ∈ 𝑧 → ((∃𝑤 𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
35	33, 34	syl 14	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → ((∃𝑤 𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)))
36	35	imp 124	. . . 4 ⊢ ((𝑧 ∈ 𝐶 ∧ (∃𝑤 𝑤 ∈ 𝑧 → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))) → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
37	3, 36	sylan2br 288	. . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
38	37	ralimiaa 2606	. 2 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → ∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
39	10, 25, 4	acexmidlem1 6054	. 2 ⊢ (∀𝑧 ∈ 𝐶 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))
40	38, 39	syl 14	1 ⊢ (∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢) → (𝜑 ∨ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ∃!wreu 2524  {crab 2526  ∅c0 3512  {csn 3694  {cpr 3695

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-iota 5317  df-riota 6011

This theorem is referenced by:  acexmidlemv  6056