Theorem safesnsupfilb 42909

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 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑅 Or 𝐴)
3		safesnsupfilb.subset	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
4	3	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ⊆ 𝐴)
5		safesnsupfilb.finite	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Fin)
6	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin)
7		simpr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8		eqidd 2726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
9	2, 4, 6, 7, 8	supgtoreq 9488	. . . . 5 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑅sup(𝐵, 𝐴, 𝑅) ∨ 𝑥 = sup(𝐵, 𝐴, 𝑅)))
10		df-or 846	. . . . . 6 ⊢ ((𝑥 = sup(𝐵, 𝐴, 𝑅) ∨ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)) ↔ (¬ 𝑥 = sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
11		orcom 868	. . . . . 6 ⊢ ((𝑥𝑅sup(𝐵, 𝐴, 𝑅) ∨ 𝑥 = sup(𝐵, 𝐴, 𝑅)) ↔ (𝑥 = sup(𝐵, 𝐴, 𝑅) ∨ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
12		df-ne 2931	. . . . . . 7 ⊢ (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) ↔ ¬ 𝑥 = sup(𝐵, 𝐴, 𝑅))
13	12	imbi1i 348	. . . . . 6 ⊢ ((𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)) ↔ (¬ 𝑥 = sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
14	10, 11, 13	3bitr4i 302	. . . . 5 ⊢ ((𝑥𝑅sup(𝐵, 𝐴, 𝑅) ∨ 𝑥 = sup(𝐵, 𝐴, 𝑅)) ↔ (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
15	9, 14	sylib 217	. . . 4 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
16	15	ralrimiva 3136	. . 3 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → ∀𝑥 ∈ 𝐵 (𝑥 ≠ sup(𝐵, 𝐴, 𝑅) → 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
17		iftrue 4531	. . . . . . 7 ⊢ (𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
18	17	difeq2d 4115	. . . . . 6 ⊢ (𝑂 ≺ 𝐵 → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)}))
19	18	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ {sup(𝐵, 𝐴, 𝑅)}))
20	19	raleqdv 3315	. . . 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 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ≺ 𝐵)
22	21	iftrued 4533	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
23	22	raleqdv 3315	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ ∀𝑦 ∈ {sup(𝐵, 𝐴, 𝑅)}𝑥𝑅𝑦))
24	5	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ∈ Fin)
25		safesnsupfilb.small	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
26	25	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝑂 = ∅ ∨ 𝑂 = 1o))
27		0elon 6419	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ On
28		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
29	27, 28	mpbiri 257	. . . . . . . . . . . . 13 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
30		1on 8492	. . . . . . . . . . . . . 14 ⊢ 1o ∈ On
31		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
32	30, 31	mpbiri 257	. . . . . . . . . . . . 13 ⊢ (𝑂 = 1o → 𝑂 ∈ On)
33	29, 32	jaoi 855	. . . . . . . . . . . 12 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
34	26, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ∈ On)
35	21, 34	sdomne0d 42905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ≠ ∅)
36	3	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ⊆ 𝐴)
37	24, 35, 36	3jca 1125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴))
38		fisupcl 9487	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
39	1, 37, 38	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
40		breq2 5148	. . . . . . . . 9 ⊢ (𝑦 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
41	40	ralsng 4674	. . . . . . . 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 278	. . . . . 6 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦 ↔ 𝑥𝑅sup(𝐵, 𝐴, 𝑅)))
44	43	ralbidv 3168	. . . . 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 4796	. . . . 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 288	. . . 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 278	. . 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 256	. 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 4509	. . 3 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ if (𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)𝑥𝑅𝑦
50		iffalse 4534	. . . . . . 7 ⊢ (¬ 𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
51	50	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
52	51	difeq2d 4115	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = (𝐵 ∖ 𝐵))
53		difid 4367	. . . . 5 ⊢ (𝐵 ∖ 𝐵) = ∅
54	52, 53	eqtrdi 2781	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → (𝐵 ∖ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)) = ∅)
55	54	raleqdv 3315	. . 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 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)
57	48, 56	pm2.61dan 811	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 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051   ∖ cdif 3938   ⊆ wss 3941  ∅c0 4319  ifcif 4525  {csn 4625   class class class wbr 5144   Or wor 5584  Oncon0 6365  1_oc1o 8473   ≺ csdm 8956  Fincfn 8957  supcsup 9458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-om 7866  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460

This theorem is referenced by: (None)