Theorem reg3exmidlemwe 4701

Description: Lemma for reg3exmid 4702. 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 4696	. 2 ⊢ E Fr 𝐴
2		epel 4413	. . . . . 6 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
3		epel 4413	. . . . . 6 ⊢ (𝑏 E 𝑐 ↔ 𝑏 ∈ 𝑐)
4	2, 3	anbi12i 460	. . . . 5 ⊢ ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) ↔ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
5		simpr 110	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
6		elirr 4663	. . . . . . . 8 ⊢ ¬ {∅} ∈ {∅}
7		simprr 533	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 ∈ 𝑐)
8		noel 3512	. . . . . . . . . . . . 13 ⊢ ¬ 𝑎 ∈ ∅
9		eleq2 2296	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ∅))
10	8, 9	mtbiri 682	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ¬ 𝑎 ∈ 𝑏)
11		simprl 531	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 ∈ 𝑏)
12	10, 11	nsyl3 631	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑏 = ∅)
13		elrabi 2970	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
15	13, 14	eleq2s 2327	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐴 → 𝑏 ∈ {∅, {∅}})
16		elpri 3712	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
17	15, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
18	17	orcomd 737	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19	18	3ad2ant2 1046	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑏 = {∅} ∨ 𝑏 = ∅))
20	19	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑏 = {∅} ∨ 𝑏 = ∅))
21	12, 20	ecased 1386	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 = {∅})
22		noel 3512	. . . . . . . . . . . . 13 ⊢ ¬ 𝑏 ∈ ∅
23		eleq2 2296	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → (𝑏 ∈ 𝑐 ↔ 𝑏 ∈ ∅))
24	22, 23	mtbiri 682	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ¬ 𝑏 ∈ 𝑐)
25	24, 7	nsyl3 631	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑐 = ∅)
26		elrabi 2970	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
27	26, 14	eleq2s 2327	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ {∅, {∅}})
28		vex 2816	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
29	28	elpr 3710	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
30	27, 29	sylib 122	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
31	30	orcomd 737	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32	31	3ad2ant3 1047	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝑐 = {∅} ∨ 𝑐 = ∅))
33	32	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → (𝑐 = {∅} ∨ 𝑐 = ∅))
34	25, 33	ecased 1386	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑐 = {∅})
35	7, 21, 34	3eltr3d 2315	. . . . . . . . 9 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → {∅} ∈ {∅})
36	35	ex 115	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐) → {∅} ∈ {∅}))
37	6, 36	mtoi 670	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ¬ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
38	37	adantr 276	. . . . . 6 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐))
39	5, 38	pm2.21dd 625	. . . . 5 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 E 𝑐)
40	4, 39	sylan2b 287	. . . 4 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 E 𝑏 ∧ 𝑏 E 𝑐)) → 𝑎 E 𝑐)
41	40	ex 115	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐))
42	41	rgen3 2629	. 2 ⊢ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐)
43		df-wetr 4455	. 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 951	1 ⊢ E We 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  {crab 2524  ∅c0 3508  {csn 3689  {cpr 3690   class class class wbr 4109   E cep 4408   Fr wfr 4449   We wwe 4451

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-eprel 4410  df-frfor 4452  df-frind 4453  df-wetr 4455

This theorem is referenced by:  reg3exmid  4702