Theorem reg3exmidlemwe 4321

Description: Lemma for reg3exmid 4322. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)

Hypothesis

Ref	Expression
reg3exmidlemwe.a	⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}

Assertion

Ref	Expression
reg3exmidlemwe	⊢ E We 𝐴

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem reg3exmidlemwe

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfregfr 4316	. 2 ⊢ E Fr 𝐴
2		epel 4047	. . . . . 6 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
3		epel 4047	. . . . . 6 ⊢ (𝑏 E 𝑐 ↔ 𝑏 ∈ 𝑐)
4	2, 3	anbi12i 447	. . . . 5 ⊢ ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) ↔ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
5		simpr 108	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
6		elirr 4284	. . . . . . . 8 ⊢ ¬ {∅} ∈ {∅}
7		simprr 498	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 ∈ 𝑐)
8		noel 3255	. . . . . . . . . . . . 13 ⊢ ¬ 𝑎 ∈ ∅
9		eleq2 2142	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ∅))
10	8, 9	mtbiri 632	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ¬ 𝑎 ∈ 𝑏)
11		simprl 497	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 ∈ 𝑏)
12	10, 11	nsyl3 588	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑏 = ∅)
13		elrabi 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
15	13, 14	eleq2s 2173	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐴 → 𝑏 ∈ {∅, {∅}})
16		elpri 3421	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
17	15, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
18	17	orcomd 680	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19	18	3ad2ant2 960	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑏 = {∅} ∨ 𝑏 = ∅))
20	19	adantr 270	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑏 = {∅} ∨ 𝑏 = ∅))
21	12, 20	ecased 1280	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 = {∅})
22		noel 3255	. . . . . . . . . . . . 13 ⊢ ¬ 𝑏 ∈ ∅
23		eleq2 2142	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → (𝑏 ∈ 𝑐 ↔ 𝑏 ∈ ∅))
24	22, 23	mtbiri 632	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ¬ 𝑏 ∈ 𝑐)
25	24, 7	nsyl3 588	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑐 = ∅)
26		elrabi 2746	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
27	26, 14	eleq2s 2173	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ {∅, {∅}})
28		vex 2604	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
29	28	elpr 3419	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
30	27, 29	sylib 120	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
31	30	orcomd 680	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32	31	3ad2ant3 961	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑐 = {∅} ∨ 𝑐 = ∅))
33	32	adantr 270	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑐 = {∅} ∨ 𝑐 = ∅))
34	25, 33	ecased 1280	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑐 = {∅})
35	7, 21, 34	3eltr3d 2161	. . . . . . . . 9 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → {∅} ∈ {∅})
36	35	ex 113	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐) → {∅} ∈ {∅}))
37	6, 36	mtoi 622	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ¬ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
38	37	adantr 270	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
39	5, 38	pm2.21dd 582	. . . . 5 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 E 𝑐)
40	4, 39	sylan2b 281	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 E 𝑏 ∧ 𝑏 E 𝑐)) → 𝑎 E 𝑐)
41	40	ex 113	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐))
42	41	rgen3 2448	. 2 ⊢ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐)
43		df-wetr 4089	. 2 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐)))
44	1, 42, 43	mpbir2an 883	1 ⊢ E We 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348  {crab 2352  ∅c0 3251  {csn 3398  {cpr 3399   class class class wbr 3785   E cep 4042   Fr wfr 4083   We wwe 4085

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-frfor 4086  df-frind 4087  df-wetr 4089

This theorem is referenced by:  reg3exmid  4322