Theorem safesnsupfilb 43442

Description: If 𝐵 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 3-Sep-2024.)

Hypotheses

Ref	Expression
safesnsupfilb.small	⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
safesnsupfilb.finite	⊢ (𝜑 → 𝐵 ∈ Fin)
safesnsupfilb.subset	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
safesnsupfilb.ordered	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

Ref	Expression
safesnsupfilb	⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑂,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem safesnsupfilb

Step	Hyp	Ref	Expression
1		safesnsupfilb.ordered	. . . . . . 7 ⊢ (𝜑 → 𝑅 Or 𝐴)
2	1	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑅 Or 𝐴)
3		safesnsupfilb.subset	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
4	3	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ⊆ 𝐴)
5		safesnsupfilb.finite	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
6	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin)
7		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8		eqidd 2736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
9	2, 4, 6, 7, 8	supgtoreq 9483	. . . . 5 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑅sup(𝐵, 𝐴, 𝑅) ∨ 𝑥 = sup(𝐵, 𝐴, 𝑅)))
10		df-or 848	. . . . . 6 ⊢ ((𝑥 = sup(𝐵, 𝐴, 𝑅) ∨ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)) ↔ (¬ 𝑥 = sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
11		orcom 870	. . . . . 6 ⊢ ((𝑥𝑅sup(𝐵, 𝐴, 𝑅) ∨ 𝑥 = sup(𝐵, 𝐴, 𝑅)) ↔ (𝑥 = sup(𝐵, 𝐴, 𝑅) ∨ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
12		df-ne 2933	. . . . . . 7 ⊢ (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) ↔ ¬ 𝑥 = sup(𝐵, 𝐴, 𝑅))
13	12	imbi1i 349	. . . . . 6 ⊢ ((𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)) ↔ (¬ 𝑥 = sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
14	10, 11, 13	3bitr4i 303	. . . . 5 ⊢ ((𝑥𝑅sup(𝐵, 𝐴, 𝑅) ∨ 𝑥 = sup(𝐵, 𝐴, 𝑅)) ↔ (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
15	9, 14	sylib 218	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
16	15	ralrimiva 3132	. . 3 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → ∀𝑥 ∈ 𝐵 (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
17		iftrue 4506	. . . . . . 7 ⊢ (𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
18	17	difeq2d 4101	. . . . . 6 ⊢ (𝑂 ≺ 𝐵 → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)}))
19	18	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)}))
20	19	raleqdv 3305	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑥 ∈ (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)})∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦))
21		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ≺ 𝐵)
22	21	iftrued 4508	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
23	22	raleqdv 3305	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑦 ∈ {sup(𝐵, 𝐴, 𝑅)}𝑥𝑅𝑦))
24	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ∈ Fin)
25		safesnsupfilb.small	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
26	25	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝑂 = ∅ ∨ 𝑂 = 1o))
27		0elon 6407	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ On
28		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
29	27, 28	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
30		1on 8492	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
31		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
32	30, 31	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑂 = 1o → 𝑂 ∈ On)
33	29, 32	jaoi 857	. . . . . . . . . . . 12 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
34	26, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ∈ On)
35	21, 34	sdomne0d 43438	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ≠ ∅)
36	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ⊆ 𝐴)
37	24, 35, 36	3jca 1128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴))
38		fisupcl 9482	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
39	1, 37, 38	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
40		breq2 5123	. . . . . . . . 9 ⊢ (𝑦 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
41	40	ralsng 4651	. . . . . . . 8 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐵 → (∀𝑦 ∈ {sup(𝐵, 𝐴, 𝑅)}𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
42	39, 41	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑦 ∈ {sup(𝐵, 𝐴, 𝑅)}𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
43	23, 42	bitrd 279	. . . . . 6 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
44	43	ralbidv 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑥 ∈ (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)})∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑥 ∈ (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)})𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
45		raldifsnb 4772	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ∀𝑥 ∈ (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)})𝑥𝑅sup(𝐵, 𝐴, 𝑅))
46	44, 45	bitr4di 289	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑥 ∈ (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)})∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅))))
47	20, 46	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅))))
48	16, 47	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → ∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦)
49		ral0 4488	. . 3 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦
50		iffalse 4509	. . . . . . 7 ⊢ (¬ 𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
51	50	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
52	51	difeq2d 4101	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ 𝐵))
53		difid 4351	. . . . 5 ⊢ (𝐵 ∖ 𝐵) = ∅
54	52, 53	eqtrdi 2786	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = ∅)
55	54	raleqdv 3305	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦))
56	49, 55	mpbiri 258	. 2 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → ∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦)
57	48, 56	pm2.61dan 812	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵))∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ifcif 4500  {csn 4601   class class class wbr 5119   Or wor 5560  Oncon0 6352  1_oc1o 8473   ≺ csdm 8958  Fincfn 8959  supcsup 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-om 7862  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454

This theorem is referenced by: (None)