Theorem reg3exmidlemwe 4556

Description: Lemma for reg3exmid 4557. 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 4551	. 2 ⊢ E Fr 𝐴
2		epel 4270	. . . . . 6 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
3		epel 4270	. . . . . 6 ⊢ (𝑏 E 𝑐 ↔ 𝑏 ∈ 𝑐)
4	2, 3	anbi12i 456	. . . . 5 ⊢ ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) ↔ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
5		simpr 109	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
6		elirr 4518	. . . . . . . 8 ⊢ ¬ {∅} ∈ {∅}
7		simprr 522	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 ∈ 𝑐)
8		noel 3413	. . . . . . . . . . . . 13 ⊢ ¬ 𝑎 ∈ ∅
9		eleq2 2230	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ∅))
10	8, 9	mtbiri 665	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ¬ 𝑎 ∈ 𝑏)
11		simprl 521	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 ∈ 𝑏)
12	10, 11	nsyl3 616	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑏 = ∅)
13		elrabi 2879	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
15	13, 14	eleq2s 2261	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐴 → 𝑏 ∈ {∅, {∅}})
16		elpri 3599	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
17	15, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
18	17	orcomd 719	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19	18	3ad2ant2 1009	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑏 = {∅} ∨ 𝑏 = ∅))
20	19	adantr 274	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑏 = {∅} ∨ 𝑏 = ∅))
21	12, 20	ecased 1339	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 = {∅})
22		noel 3413	. . . . . . . . . . . . 13 ⊢ ¬ 𝑏 ∈ ∅
23		eleq2 2230	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → (𝑏 ∈ 𝑐 ↔ 𝑏 ∈ ∅))
24	22, 23	mtbiri 665	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ¬ 𝑏 ∈ 𝑐)
25	24, 7	nsyl3 616	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑐 = ∅)
26		elrabi 2879	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
27	26, 14	eleq2s 2261	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ {∅, {∅}})
28		vex 2729	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
29	28	elpr 3597	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
30	27, 29	sylib 121	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
31	30	orcomd 719	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32	31	3ad2ant3 1010	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑐 = {∅} ∨ 𝑐 = ∅))
33	32	adantr 274	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑐 = {∅} ∨ 𝑐 = ∅))
34	25, 33	ecased 1339	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑐 = {∅})
35	7, 21, 34	3eltr3d 2249	. . . . . . . . 9 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → {∅} ∈ {∅})
36	35	ex 114	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐) → {∅} ∈ {∅}))
37	6, 36	mtoi 654	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ¬ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
38	37	adantr 274	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
39	5, 38	pm2.21dd 610	. . . . 5 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 E 𝑐)
40	4, 39	sylan2b 285	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 E 𝑏 ∧ 𝑏 E 𝑐)) → 𝑎 E 𝑐)
41	40	ex 114	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐))
42	41	rgen3 2553	. 2 ⊢ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐)
43		df-wetr 4312	. 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 932	1 ⊢ E We 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {crab 2448  ∅c0 3409  {csn 3576  {cpr 3577   class class class wbr 3982   E cep 4265   Fr wfr 4306   We wwe 4308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267  df-frfor 4309  df-frind 4310  df-wetr 4312

This theorem is referenced by:  reg3exmid  4557