Theorem acexmidlem2 5915

Description: Lemma for acexmid 5917. This builds on acexmidlem1 5914 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 5256 sense because it uses ordered pairs as described in opthreg 4588 rather than df-op 3627).

The set 𝐴 is also found in onsucelsucexmidlem 4561.

(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 2477	. . . . 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 1894	. . . . 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 2260	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ {𝐴, 𝐵})
6		vex 2763	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7	6	elpr 3639	. . . . . . . 8 ⊢ (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
8	5, 7	bitri 184	. . . . . . 7 ⊢ (𝑧 ∈ 𝐶 ↔ (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
9		onsucelsucexmidlem1 4560	. . . . . . . . . . 11 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10		acexmidlem.a	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
11	9, 10	eleqtrri 2269	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
12		elex2 2776	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ 𝐴
14		eleq2 2257	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐴))
15	14	exbidv 1836	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝐴))
16	13, 15	mpbiri 168	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ∃𝑤 𝑤 ∈ 𝑧)
17		p0ex 4217	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
18	17	prid2 3725	. . . . . . . . . . . 12 ⊢ {∅} ∈ {∅, {∅}}
19		eqid 2193	. . . . . . . . . . . . 13 ⊢ {∅} = {∅}
20	19	orci 732	. . . . . . . . . . . 12 ⊢ ({∅} = {∅} ∨ 𝜑)
21		eqeq1 2200	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
22	21	orbi1d 792	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
23	22	elrab 2916	. . . . . . . . . . . 12 ⊢ ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
24	18, 20, 23	mpbir2an 944	. . . . . . . . . . 11 ⊢ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25		acexmidlem.b	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
26	24, 25	eleqtrri 2269	. . . . . . . . . 10 ⊢ {∅} ∈ 𝐵
27		elex2 2776	. . . . . . . . . 10 ⊢ ({∅} ∈ 𝐵 → ∃𝑤 𝑤 ∈ 𝐵)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ 𝐵
29		eleq2 2257	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵))
30	29	exbidv 1836	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝐵))
31	28, 30	mpbiri 168	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ∃𝑤 𝑤 ∈ 𝑧)
32	16, 31	jaoi 717	. . . . . . 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 2556	. 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 5914	. 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 709  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∃!wreu 2474  {crab 2476  ∅c0 3446  {csn 3618  {cpr 3619

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-iota 5215  df-riota 5873

This theorem is referenced by:  acexmidlemv  5916