Theorem reg3exmidlemwe 4357

Description: Lemma for reg3exmid 4358. 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 4352	. 2 ⊢ E Fr 𝐴
2		epel 4083	. . . . . 6 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
3		epel 4083	. . . . . 6 ⊢ (𝑏 E 𝑐 ↔ 𝑏 ∈ 𝑐)
4	2, 3	anbi12i 448	. . . . 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 4320	. . . . . . . 8 ⊢ ¬ {∅} ∈ {∅}
7		simprr 499	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 ∈ 𝑐)
8		noel 3273	. . . . . . . . . . . . 13 ⊢ ¬ 𝑎 ∈ ∅
9		eleq2 2146	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ∅))
10	8, 9	mtbiri 633	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ¬ 𝑎 ∈ 𝑏)
11		simprl 498	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 ∈ 𝑏)
12	10, 11	nsyl3 589	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑏 = ∅)
13		elrabi 2756	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
15	13, 14	eleq2s 2177	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐴 → 𝑏 ∈ {∅, {∅}})
16		elpri 3445	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
17	15, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
18	17	orcomd 681	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19	18	3ad2ant2 961	. . . . . . . . . . . 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 1281	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 = {∅})
22		noel 3273	. . . . . . . . . . . . 13 ⊢ ¬ 𝑏 ∈ ∅
23		eleq2 2146	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → (𝑏 ∈ 𝑐 ↔ 𝑏 ∈ ∅))
24	22, 23	mtbiri 633	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ¬ 𝑏 ∈ 𝑐)
25	24, 7	nsyl3 589	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑐 = ∅)
26		elrabi 2756	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
27	26, 14	eleq2s 2177	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ {∅, {∅}})
28		vex 2615	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
29	28	elpr 3443	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
30	27, 29	sylib 120	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
31	30	orcomd 681	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32	31	3ad2ant3 962	. . . . . . . . . . . 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 1281	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑐 = {∅})
35	7, 21, 34	3eltr3d 2165	. . . . . . . . 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 623	. . . . . . 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 583	. . . . 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 2454	. 2 ⊢ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐)
43		df-wetr 4125	. 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 884	1 ⊢ E We 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 662   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ∀wral 2353  {crab 2357  ∅c0 3269  {csn 3422  {cpr 3423   class class class wbr 3811   E cep 4078   Fr wfr 4119   We wwe 4121

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-setind 4316

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-eprel 4080  df-frfor 4122  df-frind 4123  df-wetr 4125

This theorem is referenced by:  reg3exmid  4358