Theorem exmidontriimlem4 7570

Description: Lemma for exmidontriim 7571. The induction step for the induction on 𝐴. (Contributed by Jim Kingdon, 10-Aug-2024.)

Hypotheses

Ref	Expression
exmidontriimlem4.a	⊢ (𝜑 → 𝐴 ∈ On)
exmidontriimlem4.b	⊢ (𝜑 → 𝐵 ∈ On)
exmidontriimlem4.em	⊢ (𝜑 → EXMID)
exmidontriimlem4.h	⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))

Assertion

Ref	Expression
exmidontriimlem4	⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Distinct variable group:   𝑦,𝐴,𝑧

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

Proof of Theorem exmidontriimlem4

Dummy variables 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2302	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2248	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 = 𝑏 ↔ 𝐴 = 𝐵))
3		eleq1 2301	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	1, 2, 3	3orbi123d 1352	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
5		eleq2w 2300	. . . . . . 7 ⊢ (𝑏 = 𝑤 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝑤))
6		eqeq2 2248	. . . . . . 7 ⊢ (𝑏 = 𝑤 → (𝐴 = 𝑏 ↔ 𝐴 = 𝑤))
7		eleq1w 2299	. . . . . . 7 ⊢ (𝑏 = 𝑤 → (𝑏 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
8	5, 6, 7	3orbi123d 1352	. . . . . 6 ⊢ (𝑏 = 𝑤 → ((𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴) ↔ (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴)))
9	8	imbi2d 230	. . . . 5 ⊢ (𝑏 = 𝑤 → ((𝜑 → (𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴)) ↔ (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))))
10		exmidontriimlem4.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ On)
11	10	adantl 277	. . . . . . 7 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → 𝐴 ∈ On)
12		simpll 531	. . . . . . 7 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → 𝑏 ∈ On)
13		exmidontriimlem4.em	. . . . . . . 8 ⊢ (𝜑 → EXMID)
14	13	adantl 277	. . . . . . 7 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → EXMID)
15		exmidontriimlem4.h	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))
16	15	adantl 277	. . . . . . 7 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ On (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦 ∨ 𝑦 ∈ 𝑧))
17		simplr 533	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) ∧ 𝑣 ∈ 𝑏) → 𝜑)
18		eleq2w 2300	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑣 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑣))
19		eqeq2 2248	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑣 → (𝐴 = 𝑤 ↔ 𝐴 = 𝑣))
20		eleq1w 2299	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
21	18, 19, 20	3orbi123d 1352	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑣 → ((𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴) ↔ (𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴)))
22	21	imbi2d 230	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ((𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴)) ↔ (𝜑 → (𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴))))
23		simpllr 540	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) ∧ 𝑣 ∈ 𝑏) → ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴)))
24		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) ∧ 𝑣 ∈ 𝑏) → 𝑣 ∈ 𝑏)
25	22, 23, 24	rspcdva 2934	. . . . . . . . . 10 ⊢ ((((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) ∧ 𝑣 ∈ 𝑏) → (𝜑 → (𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴)))
26	17, 25	mpd 13	. . . . . . . . 9 ⊢ ((((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) ∧ 𝑣 ∈ 𝑏) → (𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴))
27	26	ralrimiva 2623	. . . . . . . 8 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → ∀𝑣 ∈ 𝑏 (𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴))
28		eleq2w 2300	. . . . . . . . . 10 ⊢ (𝑣 = 𝑦 → (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑦))
29		eqeq2 2248	. . . . . . . . . 10 ⊢ (𝑣 = 𝑦 → (𝐴 = 𝑣 ↔ 𝐴 = 𝑦))
30		eleq1w 2299	. . . . . . . . . 10 ⊢ (𝑣 = 𝑦 → (𝑣 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
31	28, 29, 30	3orbi123d 1352	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ((𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴)))
32	31	cbvralv 2786	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝑏 (𝐴 ∈ 𝑣 ∨ 𝐴 = 𝑣 ∨ 𝑣 ∈ 𝐴) ↔ ∀𝑦 ∈ 𝑏 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))
33	27, 32	sylib 122	. . . . . . 7 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → ∀𝑦 ∈ 𝑏 (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦 ∨ 𝑦 ∈ 𝐴))
34	11, 12, 14, 16, 33	exmidontriimlem3 7569	. . . . . 6 ⊢ (((𝑏 ∈ On ∧ ∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴))) ∧ 𝜑) → (𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴))
35	34	exp31 364	. . . . 5 ⊢ (𝑏 ∈ On → (∀𝑤 ∈ 𝑏 (𝜑 → (𝐴 ∈ 𝑤 ∨ 𝐴 = 𝑤 ∨ 𝑤 ∈ 𝐴)) → (𝜑 → (𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴))))
36	9, 35	tfis2 4727	. . . 4 ⊢ (𝑏 ∈ On → (𝜑 → (𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴)))
37	36	impcom 125	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ On) → (𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴))
38	37	ralrimiva 2623	. 2 ⊢ (𝜑 → ∀𝑏 ∈ On (𝐴 ∈ 𝑏 ∨ 𝐴 = 𝑏 ∨ 𝑏 ∈ 𝐴))
39		exmidontriimlem4.b	. 2 ⊢ (𝜑 → 𝐵 ∈ On)
40	4, 38, 39	rspcdva 2934	1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∨ w3o 1008   = wceq 1402   ∈ wcel 2209  ∀wral 2528  EXMIDwem 4326  Oncon0 4503

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 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 4244  ax-nul 4254  ax-pow 4306  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  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 3687  df-sn 3711  df-uni 3931  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508

This theorem is referenced by:  exmidontriim  7571