Theorem acexmidlem2 5509

Description: Lemma for acexmid 5511. This builds on acexmidlem1 5508 by noting that every element of 𝐶 is inhabited.

(Note that 𝑦 is not quite a function in the df-fun 4904 sense because it uses ordered pairs as described in opthreg 4280 rather than df-op 3384).

The set 𝐴 is also found in onsucelsucexmidlem 4254.

(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 2311	. . . . 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 1763	. . . . 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 174	. . . 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 2104	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ {𝐴, 𝐵})
6		vex 2560	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7	6	elpr 3396	. . . . . . . 8 ⊢ (𝑧 ∈ {𝐴, 𝐵} ↔ (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
8	5, 7	bitri 173	. . . . . . 7 ⊢ (𝑧 ∈ 𝐶 ↔ (𝑧 = 𝐴 ∨ 𝑧 = 𝐵))
9		onsucelsucexmidlem1 4253	. . . . . . . . . . 11 ⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
10		acexmidlem.a	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
11	9, 10	eleqtrri 2113	. . . . . . . . . 10 ⊢ ∅ ∈ 𝐴
12		elex2 2570	. . . . . . . . . 10 ⊢ (∅ ∈ 𝐴 → ∃𝑤 𝑤 ∈ 𝐴)
13	11, 12	ax-mp 7	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ 𝐴
14		eleq2 2101	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐴))
15	14	exbidv 1706	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝐴))
16	13, 15	mpbiri 157	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ∃𝑤 𝑤 ∈ 𝑧)
17		p0ex 3939	. . . . . . . . . . . . 13 ⊢ {∅} ∈ V
18	17	prid2 3477	. . . . . . . . . . . 12 ⊢ {∅} ∈ {∅, {∅}}
19		eqid 2040	. . . . . . . . . . . . 13 ⊢ {∅} = {∅}
20	19	orci 650	. . . . . . . . . . . 12 ⊢ ({∅} = {∅} ∨ 𝜑)
21		eqeq1 2046	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅}))
22	21	orbi1d 705	. . . . . . . . . . . . 13 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ 𝜑) ↔ ({∅} = {∅} ∨ 𝜑)))
23	22	elrab 2698	. . . . . . . . . . . 12 ⊢ ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ 𝜑)))
24	18, 20, 23	mpbir2an 849	. . . . . . . . . . 11 ⊢ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
25		acexmidlem.b	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
26	24, 25	eleqtrri 2113	. . . . . . . . . 10 ⊢ {∅} ∈ 𝐵
27		elex2 2570	. . . . . . . . . 10 ⊢ ({∅} ∈ 𝐵 → ∃𝑤 𝑤 ∈ 𝐵)
28	26, 27	ax-mp 7	. . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ 𝐵
29		eleq2 2101	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵))
30	29	exbidv 1706	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (∃𝑤 𝑤 ∈ 𝑧 ↔ ∃𝑤 𝑤 ∈ 𝐵))
31	28, 30	mpbiri 157	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → ∃𝑤 𝑤 ∈ 𝑧)
32	16, 31	jaoi 636	. . . . . . 7 ⊢ ((𝑧 = 𝐴 ∨ 𝑧 = 𝐵) → ∃𝑤 𝑤 ∈ 𝑧)
33	8, 32	sylbi 114	. . . . . 6 ⊢ (𝑧 ∈ 𝐶 → ∃𝑤 𝑤 ∈ 𝑧)
34		pm2.27 35	. . . . . 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 115	. . . 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 272	. . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ ∀𝑤 ∈ 𝑧 ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢)) → ∃!𝑣 ∈ 𝑧 ∃𝑢 ∈ 𝑦 (𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢))
38	37	ralimiaa 2383	. 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 5508	. 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 97   ∨ wo 629  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  ∃!wreu 2308  {crab 2310  ∅c0 3224  {csn 3375  {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-iota 4867  df-riota 5468

This theorem is referenced by:  acexmidlemv  5510