Theorem fimax2gtrilemstep 6895

Description: Lemma for fimax2gtri 6896. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)

Hypotheses

Ref	Expression
fimax2gtri.po	⊢ (𝜑 → 𝑅 Po 𝐴)
fimax2gtri.tri	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
fimax2gtri.fin	⊢ (𝜑 → 𝐴 ∈ Fin)
fimax2gtri.n0	⊢ (𝜑 → 𝐴 ≠ ∅)
fimax2gtri.ufin	⊢ (𝜑 → 𝑈 ∈ Fin)
fimax2gtri.uss	⊢ (𝜑 → 𝑈 ⊆ 𝐴)
fimax2gtri.za	⊢ (𝜑 → 𝑍 ∈ 𝐴)
fimax2gtri.va	⊢ (𝜑 → 𝑉 ∈ 𝐴)
fimax2gtri.vu	⊢ (𝜑 → ¬ 𝑉 ∈ 𝑈)
fimax2gtri.zb	⊢ (𝜑 → ∀𝑦 ∈ 𝑈 ¬ 𝑍𝑅𝑦)

Assertion

Ref	Expression
fimax2gtrilemstep	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝐴,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑍,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem fimax2gtrilemstep

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fimax2gtri.va	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
2		fimax2gtri.za	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
3		fimax2gtri.po	. . . 4 ⊢ (𝜑 → 𝑅 Po 𝐴)
4		fimax2gtri.tri	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
5	3, 4, 2, 1	tridc 6894	. . 3 ⊢ (𝜑 → DECID 𝑍𝑅𝑉)
6	1, 2, 5	ifcldcd 3570	. 2 ⊢ (𝜑 → if(𝑍𝑅𝑉, 𝑉, 𝑍) ∈ 𝐴)
7		simplr 528	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑍𝑅𝑉)
8		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → 𝑍𝑅𝑉)
9	8	iftrued 3541	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → if(𝑍𝑅𝑉, 𝑉, 𝑍) = 𝑉)
10	9	breq1d 4011	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ 𝑉𝑅𝑤))
11	10	biimpa 296	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑉𝑅𝑤)
12	11	adantllr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑉𝑅𝑤)
13	3	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → 𝑅 Po 𝐴)
14	2	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → 𝑍 ∈ 𝐴)
15	1	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → 𝑉 ∈ 𝐴)
16		fimax2gtri.uss	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ 𝐴)
17	16	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → 𝑈 ⊆ 𝐴)
18		simplr 528	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → 𝑤 ∈ 𝑈)
19	17, 18	sseldd 3156	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → 𝑤 ∈ 𝐴)
20		potr 4306	. . . . . . . . . 10 ⊢ ((𝑅 Po 𝐴 ∧ (𝑍 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑍𝑅𝑉 ∧ 𝑉𝑅𝑤) → 𝑍𝑅𝑤))
21	13, 14, 15, 19, 20	syl13anc 1240	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → ((𝑍𝑅𝑉 ∧ 𝑉𝑅𝑤) → 𝑍𝑅𝑤))
22	21	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → ((𝑍𝑅𝑉 ∧ 𝑉𝑅𝑤) → 𝑍𝑅𝑤))
23	7, 12, 22	mp2and 433	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑍𝑅𝑤)
24		fimax2gtri.zb	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ 𝑈 ¬ 𝑍𝑅𝑦)
25		breq2 4005	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑍𝑅𝑦 ↔ 𝑍𝑅𝑤))
26	25	notbid 667	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (¬ 𝑍𝑅𝑦 ↔ ¬ 𝑍𝑅𝑤))
27	26	cbvralv 2703	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑈 ¬ 𝑍𝑅𝑦 ↔ ∀𝑤 ∈ 𝑈 ¬ 𝑍𝑅𝑤)
28	24, 27	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑤 ∈ 𝑈 ¬ 𝑍𝑅𝑤)
29	28	r19.21bi 2565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑈) → ¬ 𝑍𝑅𝑤)
30	29	ad2antrr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → ¬ 𝑍𝑅𝑤)
31	23, 30	pm2.65da 661	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
32	29	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ ¬ 𝑍𝑅𝑉) → ¬ 𝑍𝑅𝑤)
33		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → ¬ 𝑍𝑅𝑉)
34	33	iffalsed 3544	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → if(𝑍𝑅𝑉, 𝑉, 𝑍) = 𝑍)
35	34	breq1d 4011	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ 𝑍𝑅𝑤))
36	35	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ ¬ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ 𝑍𝑅𝑤))
37	32, 36	mtbird 673	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑈) ∧ ¬ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
38		exmiddc 836	. . . . . . . 8 ⊢ (DECID 𝑍𝑅𝑉 → (𝑍𝑅𝑉 ∨ ¬ 𝑍𝑅𝑉))
39	5, 38	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑍𝑅𝑉 ∨ ¬ 𝑍𝑅𝑉))
40	39	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑈) → (𝑍𝑅𝑉 ∨ ¬ 𝑍𝑅𝑉))
41	31, 37, 40	mpjaodan 798	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑈) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
42	41	ralrimiva 2550	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
43		breq2 4005	. . . . . 6 ⊢ (𝑤 = 𝑦 → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
44	43	notbid 667	. . . . 5 ⊢ (𝑤 = 𝑦 → (¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
45	44	cbvralv 2703	. . . 4 ⊢ (∀𝑤 ∈ 𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ ∀𝑦 ∈ 𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
46	42, 45	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
47	3	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → 𝑅 Po 𝐴)
48	1	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → 𝑉 ∈ 𝐴)
49		poirr 4305	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝑉 ∈ 𝐴) → ¬ 𝑉𝑅𝑉)
50	47, 48, 49	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → ¬ 𝑉𝑅𝑉)
51	9	breq1d 4011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉 ↔ 𝑉𝑅𝑉))
52	50, 51	mtbird 673	. . . . 5 ⊢ ((𝜑 ∧ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉)
53	34	breq1d 4011	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉 ↔ 𝑍𝑅𝑉))
54	33, 53	mtbird 673	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉)
55	52, 54, 39	mpjaodan 798	. . . 4 ⊢ (𝜑 → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉)
56		breq2 4005	. . . . . . 7 ⊢ (𝑦 = 𝑉 → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
57	56	notbid 667	. . . . . 6 ⊢ (𝑦 = 𝑉 → (¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
58	57	ralsng 3632	. . . . 5 ⊢ (𝑉 ∈ 𝐴 → (∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
59	1, 58	syl 14	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
60	55, 59	mpbird 167	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
61		ralun 3317	. . 3 ⊢ ((∀𝑦 ∈ 𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ∧ ∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦) → ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
62	46, 60, 61	syl2anc 411	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
63		breq1 4004	. . . . 5 ⊢ (𝑥 = if(𝑍𝑅𝑉, 𝑉, 𝑍) → (𝑥𝑅𝑦 ↔ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
64	63	notbid 667	. . . 4 ⊢ (𝑥 = if(𝑍𝑅𝑉, 𝑉, 𝑍) → (¬ 𝑥𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
65	64	ralbidv 2477	. . 3 ⊢ (𝑥 = if(𝑍𝑅𝑉, 𝑉, 𝑍) → (∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
66	65	rspcev 2841	. 2 ⊢ ((if(𝑍𝑅𝑉, 𝑉, 𝑍) ∈ 𝐴 ∧ ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)
67	6, 62, 66	syl2anc 411	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   ∨ w3o 977   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  ifcif 3534  {csn 3592   class class class wbr 4001   Po wpo 4292  Fincfn 6735

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002  df-po 4294

This theorem is referenced by:  fimax2gtri  6896