Theorem safesnsupfiss 44003

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, 1-Sep-2024.)

Hypotheses

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

Assertion

Ref	Expression
safesnsupfiss	⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵)

Proof of Theorem safesnsupfiss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elif 4527	. . 3 ⊢ (𝑥 ∈ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ↔ ((𝑂 ≺ 𝐵 ∧ 𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)}) ∨ (¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵)))
2		elsni 4602	. . . . . 6 ⊢ (𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)} → 𝑥 = sup(𝐵, 𝐴, 𝑅))
3		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 = sup(𝐵, 𝐴, 𝑅)) → 𝑥 = sup(𝐵, 𝐴, 𝑅))
4		safesnsupfiss.ordered	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 Or 𝐴)
5	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑅 Or 𝐴)
6		safesnsupfiss.finite	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ Fin)
7	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ∈ Fin)
8		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ≺ 𝐵)
9		safesnsupfiss.small	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
10		0elon 6405	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ On
11		eleq1 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
12	10, 11	mpbiri 261	. . . . . . . . . . . . . 14 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
13		1on 8454	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ On
14		eleq1 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
15	13, 14	mpbiri 261	. . . . . . . . . . . . . 14 ⊢ (𝑂 = 1o → 𝑂 ∈ On)
16	12, 15	jaoi 870	. . . . . . . . . . . . 13 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
17	9, 16	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ On)
18	17	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝑂 ∈ On)
19	8, 18	sdomne0d 44002	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ≠ ∅)
20		safesnsupfiss.subset	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
21	20	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → 𝐵 ⊆ 𝐴)
22		fisupcl 9418	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
23	5, 7, 19, 21, 22	syl13anc 1395	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
24	23	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 = sup(𝐵, 𝐴, 𝑅)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
25	3, 24	eqeltrd 2865	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑂 ≺ 𝐵) ∧ 𝑥 = sup(𝐵, 𝐴, 𝑅)) → 𝑥 ∈ 𝐵)
26	25	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝑥 = sup(𝐵, 𝐴, 𝑅) → 𝑥 ∈ 𝐵))
27	2, 26	syl5 35	. . . . 5 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → (𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)} → 𝑥 ∈ 𝐵))
28	27	expimpd 458	. . . 4 ⊢ (𝜑 → ((𝑂 ≺ 𝐵 ∧ 𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)}) → 𝑥 ∈ 𝐵))
29		simpr 489	. . . . 5 ⊢ ((¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → ((¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵))
31	28, 30	jaod 872	. . 3 ⊢ (𝜑 → (((𝑂 ≺ 𝐵 ∧ 𝑥 ∈ {sup(𝐵, 𝐴, 𝑅)}) ∨ (¬ 𝑂 ≺ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵))
32	1, 31	biimtrid 245	. 2 ⊢ (𝜑 → (𝑥 ∈ if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) → 𝑥 ∈ 𝐵))
33	32	ssrdv 3945	1 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ⊆ wss 3907  ∅c0 4288  ifcif 4483  {csn 4585   class class class wbr 5105   Or wor 5559  Oncon0 6350  1_oc1o 8434   ≺ csdm 8930  Fincfn 8931  supcsup 9388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-om 7851  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390

This theorem is referenced by:  safesnsupfiub  44004