Theorem exmidontriimlem3 7187

Description: Lemma for exmidontriim 7189. What we get to do based on induction on both 𝐴 and 𝐵. (Contributed by Jim Kingdon, 10-Aug-2024.)

Hypotheses

Ref	Expression
exmidontriimlem3.a	⊢ (𝜑 → 𝐴 ∈ On)
exmidontriimlem3.b	⊢ (𝜑 → 𝐵 ∈ On)
exmidontriimlem3.em	⊢ (𝜑 → EXMID)
exmidontriimlem3.ha	⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))
exmidontriimlem3.hb	⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))

Assertion

Ref	Expression
exmidontriimlem3	⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑧)

Proof of Theorem exmidontriimlem3

Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3mix1 1161	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
2	1	adantl 275	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
3		3mix3 1163	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
4	3	adantl 275	. . 3 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
5		simpr 109	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵)
6		dfss3 3137	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵)
7	5, 6	sylibr 133	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
8		simplr 525	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴)
9		dfss3 3137	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴)
10	8, 9	sylibr 133	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → 𝐵 ⊆ 𝐴)
11	7, 10	eqssd 3164	. . . 4 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → 𝐴 = 𝐵)
12	11	3mix2d 1168	. . 3 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
13		exmidontriimlem3.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
14		exmidontriimlem3.em	. . . . 5 ⊢ (𝜑 → EXMID)
15		exmidontriimlem3.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ On)
16		exmidontriimlem3.ha	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))
17		eleq1 2233	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝑦 ↔ 𝑢 ∈ 𝑦))
18		equequ1 1705	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑧 = 𝑦 ↔ 𝑢 = 𝑦))
19		eleq2 2234	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑢))
20	17, 18, 19	3orbi123d 1306	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧) ↔ (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢)))
21	20	ralbidv 2470	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧) ↔ ∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢)))
22	21	cbvralv 2696	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧) ↔ ∀𝑢 ∈ 𝐴 ∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢))
23	16, 22	sylib 121	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢))
24		eleq2 2234	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝐵))
25		eqeq2 2180	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑢 = 𝑦 ↔ 𝑢 = 𝐵))
26		eleq1 2233	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
27	24, 25, 26	3orbi123d 1306	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢) ↔ (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢)))
28	27	rspcv 2830	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢) → (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢)))
29	28	ralimdv 2538	. . . . . . 7 ⊢ (𝐵 ∈ On → (∀𝑢 ∈ 𝐴 ∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢) → ∀𝑢 ∈ 𝐴 (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢)))
30	15, 23, 29	sylc 62	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢))
31		biid 170	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 ↔ 𝑢 ∈ 𝐵)
32		eqcom 2172	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 ↔ 𝐵 = 𝑢)
33		biid 170	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢)
34	31, 32, 33	3orbi123i 1184	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢) ↔ (𝑢 ∈ 𝐵 ∨ 𝐵 = 𝑢 ∨ 𝐵 ∈ 𝑢))
35		3orcomb 982	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝐵 = 𝑢 ∨ 𝐵 ∈ 𝑢) ↔ (𝑢 ∈ 𝐵 ∨ 𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢))
36		3orrot 979	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢) ↔ (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
37	34, 35, 36	3bitri 205	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
38	37	ralbii 2476	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢) ↔ ∀𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
39	30, 38	sylib 121	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
40	13, 14, 39	exmidontriimlem2 7186	. . . 4 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∨ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵))
41	40	adantr 274	. . 3 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) → (𝐵 ∈ 𝐴 ∨ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵))
42	4, 12, 41	mpjaodan 793	. 2 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
43		exmidontriimlem3.hb	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))
44		eleq2 2234	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
45		eqeq2 2180	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝑦 ↔ 𝐴 = 𝑤))
46		eleq1 2233	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
47	44, 45, 46	3orbi123d 1306	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴)))
48	47	cbvralv 2696	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ ∀𝑤 ∈ 𝐵 (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))
49	43, 48	sylib 121	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))
50	15, 14, 49	exmidontriimlem2 7186	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴))
51	2, 42, 50	mpjaodan 793	1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 703   ∨ w3o 972   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ⊆ wss 3121  EXMIDwem 4178  Oncon0 4346

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-uni 3795  df-tr 4086  df-exmid 4179  df-iord 4349  df-on 4351

This theorem is referenced by:  exmidontriimlem4  7188