Theorem reg3exmidlemwe 4572

Description: Lemma for reg3exmid 4573. 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 4567	. 2 ⊢ E Fr 𝐴
2		epel 4286	. . . . . 6 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏)
3		epel 4286	. . . . . 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 4534	. . . . . . . 8 ⊢ ¬ {∅} ∈ {∅}
7		simprr 531	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 ∈ 𝑐)
8		noel 3424	. . . . . . . . . . . . 13 ⊢ ¬ 𝑎 ∈ ∅
9		eleq2 2239	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ∅))
10	8, 9	mtbiri 675	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ¬ 𝑎 ∈ 𝑏)
11		simprl 529	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑎 ∈ 𝑏)
12	10, 11	nsyl3 626	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑏 = ∅)
13		elrabi 2888	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑏 ∈ {∅, {∅}})
14		reg3exmidlemwe.a	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}
15	13, 14	eleq2s 2270	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ 𝐴 → 𝑏 ∈ {∅, {∅}})
16		elpri 3612	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {∅, {∅}} → (𝑏 = ∅ ∨ 𝑏 = {∅}))
17	15, 16	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = ∅ ∨ 𝑏 = {∅}))
18	17	orcomd 729	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐴 → (𝑏 = {∅} ∨ 𝑏 = ∅))
19	18	3ad2ant2 1019	. . . . . . . . . . . 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 1349	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑏 = {∅})
22		noel 3424	. . . . . . . . . . . . 13 ⊢ ¬ 𝑏 ∈ ∅
23		eleq2 2239	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → (𝑏 ∈ 𝑐 ↔ 𝑏 ∈ ∅))
24	22, 23	mtbiri 675	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ¬ 𝑏 ∈ 𝑐)
25	24, 7	nsyl3 626	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → ¬ 𝑐 = ∅)
26		elrabi 2888	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} → 𝑐 ∈ {∅, {∅}})
27	26, 14	eleq2s 2270	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ {∅, {∅}})
28		vex 2738	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
29	28	elpr 3610	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {∅, {∅}} ↔ (𝑐 = ∅ ∨ 𝑐 = {∅}))
30	27, 29	sylib 122	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = ∅ ∨ 𝑐 = {∅}))
31	30	orcomd 729	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐴 → (𝑐 = {∅} ∨ 𝑐 = ∅))
32	31	3ad2ant3 1020	. . . . . . . . . . . 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 1349	. . . . . . . . . 10 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑎 ∈ 𝑏 ∧ 𝑏 ∈ 𝑐)) → 𝑐 = {∅})
35	7, 21, 34	3eltr3d 2258	. . . . . . . . 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 664	. . . . . . 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 620	. . . . 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 2562	. 2 ⊢ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ∀𝑐 ∈ 𝐴 ((𝑎 E 𝑏 ∧ 𝑏 E 𝑐) → 𝑎 E 𝑐)
43		df-wetr 4328	. 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 942	1 ⊢ E We 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2146  ∀wral 2453  {crab 2457  ∅c0 3420  {csn 3589  {cpr 3590   class class class wbr 3998   E cep 4281   Fr wfr 4322   We wwe 4324

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-eprel 4283  df-frfor 4325  df-frind 4326  df-wetr 4328

This theorem is referenced by:  reg3exmid  4573