Theorem exmidontriimlem3 7580

Description: Lemma for exmidontriim 7582. 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 1197	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
2	1	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
3		3mix3 1199	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
4	3	adantl 277	. . 3 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
5		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵)
6		dfss3 3236	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵)
7	5, 6	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
8		simplr 533	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴)
9		dfss3 3236	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴)
10	8, 9	sylibr 134	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → 𝐵 ⊆ 𝐴)
11	7, 10	eqssd 3265	. . . 4 ⊢ (((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) ∧ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵) → 𝐴 = 𝐵)
12	11	3mix2d 1204	. . 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 2301	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑧 ∈ 𝑦 ↔ 𝑢 ∈ 𝑦))
18		equequ1 1764	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑧 = 𝑦 ↔ 𝑢 = 𝑦))
19		eleq2 2302	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑢))
20	17, 18, 19	3orbi123d 1352	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → ((𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧) ↔ (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢)))
21	20	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧) ↔ ∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢)))
22	21	cbvralv 2786	. . . . . . . 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 122	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢))
24		eleq2 2302	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝐵))
25		eqeq2 2248	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑢 = 𝑦 ↔ 𝑢 = 𝐵))
26		eleq1 2301	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢))
27	24, 25, 26	3orbi123d 1352	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢) ↔ (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢)))
28	27	rspcv 2925	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑦 ∈ On (𝑢 ∈ 𝑦 ∨ 𝑢 = 𝑦 ∨ 𝑦 ∈ 𝑢) → (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢)))
29	28	ralimdv 2618	. . . . . . 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 171	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 ↔ 𝑢 ∈ 𝐵)
32		eqcom 2240	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 ↔ 𝐵 = 𝑢)
33		biid 171	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑢 ↔ 𝐵 ∈ 𝑢)
34	31, 32, 33	3orbi123i 1220	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢) ↔ (𝑢 ∈ 𝐵 ∨ 𝐵 = 𝑢 ∨ 𝐵 ∈ 𝑢))
35		3orcomb 1018	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝐵 = 𝑢 ∨ 𝐵 ∈ 𝑢) ↔ (𝑢 ∈ 𝐵 ∨ 𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢))
36		3orrot 1015	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢) ↔ (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
37	34, 35, 36	3bitri 206	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢) ↔ (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
38	37	ralbii 2556	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 (𝑢 ∈ 𝐵 ∨ 𝑢 = 𝐵 ∨ 𝐵 ∈ 𝑢) ↔ ∀𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
39	30, 38	sylib 122	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∨ 𝐵 = 𝑢 ∨ 𝑢 ∈ 𝐵))
40	13, 14, 39	exmidontriimlem2 7579	. . . 4 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∨ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵))
41	40	adantr 276	. . 3 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴) → (𝐵 ∈ 𝐴 ∨ ∀𝑢 ∈ 𝐴 𝑢 ∈ 𝐵))
42	4, 12, 41	mpjaodan 810	. 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 2302	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑤))
45		eqeq2 2248	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝑦 ↔ 𝐴 = 𝑤))
46		eleq1 2301	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
47	44, 45, 46	3orbi123d 1352	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴)))
48	47	cbvralv 2786	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴) ↔ ∀𝑤 ∈ 𝐵 (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))
49	43, 48	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))
50	15, 14, 49	exmidontriimlem2 7579	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ ∀𝑤 ∈ 𝐵 𝑤 ∈ 𝐴))
51	2, 42, 50	mpjaodan 810	1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  EXMIDwem 4331  Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-uni 3936  df-tr 4230  df-exmid 4332  df-iord 4511  df-on 4513

This theorem is used by:  exmidontriimlem4  7581