Theorem safesnsupfilb 43516

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 2732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
9	2, 4, 6, 7, 8	supgtoreq 9361	. . . . 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 2929	. . . . . . 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 3124	. . 3 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → ∀𝑥 ∈ 𝐵 (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
17		iftrue 4480	. . . . . . 7 ⊢ (𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
18	17	difeq2d 4075	. . . . . 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 3292	. . . 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 4482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
23	22	raleqdv 3292	. . . . . . 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 6367	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ On
28		eleq1 2819	. . . . . . . . . . . . . 14 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
29	27, 28	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
30		1on 8403	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
31		eleq1 2819	. . . . . . . . . . . . . 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 43512	. . . . . . . . . 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 9360	. . . . . . . . 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 5097	. . . . . . . . 9 ⊢ (𝑦 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
41	40	ralsng 4627	. . . . . . . 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 3155	. . . . 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 4747	. . . . 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 4462	. . 3 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦
50		iffalse 4483	. . . . . . 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 4075	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ 𝐵))
53		difid 4325	. . . . 5 ⊢ (𝐵 ∖ 𝐵) = ∅
54	52, 53	eqtrdi 2782	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = ∅)
55	54	raleqdv 3292	. . 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 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4282  ifcif 4474  {csn 4575   class class class wbr 5093   Or wor 5526  Oncon0 6312  1_oc1o 8384   ≺ csdm 8874  Fincfn 8875  supcsup 9330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-om 7803  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9332

This theorem is referenced by: (None)