Theorem safesnsupfidom1o 43406

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
safesnsupfidom1o.small	⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
safesnsupfidom1o.finite	⊢ (𝜑 → 𝐵 ∈ Fin)

Assertion

Ref	Expression
safesnsupfidom1o	⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)

Proof of Theorem safesnsupfidom1o

Step	Hyp	Ref	Expression
1		iftrue 4494	. . . 4 ⊢ (𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
2	1	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
3		ensn1g 8993	. . . . 5 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4		1on 8446	. . . . . 6 ⊢ 1o ∈ On
5		domrefg 8958	. . . . . 6 ⊢ (1o ∈ On → 1o ≼ 1o)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1o ≼ 1o
7		endomtr 8983	. . . . 5 ⊢ (({sup(𝐵, 𝐴, 𝑅)} ≈ 1o ∧ 1o ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
8	3, 6, 7	sylancl 586	. . . 4 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
9		snprc 4681	. . . . . 6 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V ↔ {sup(𝐵, 𝐴, 𝑅)} = ∅)
10		snex 5391	. . . . . . 7 ⊢ {sup(𝐵, 𝐴, 𝑅)} ∈ V
11		eqeng 8957	. . . . . . 7 ⊢ ({sup(𝐵, 𝐴, 𝑅)} ∈ V → ({sup(𝐵, 𝐴, 𝑅)} = ∅ → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ ({sup(𝐵, 𝐴, 𝑅)} = ∅ → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅)
13	9, 12	sylbi 217	. . . . 5 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅)
14		0domg 9068	. . . . . 6 ⊢ (1o ∈ On → ∅ ≼ 1o)
15	4, 14	ax-mp 5	. . . . 5 ⊢ ∅ ≼ 1o
16		endomtr 8983	. . . . 5 ⊢ (({sup(𝐵, 𝐴, 𝑅)} ≈ ∅ ∧ ∅ ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
17	13, 15, 16	sylancl 586	. . . 4 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
18	8, 17	pm2.61i 182	. . 3 ⊢ {sup(𝐵, 𝐴, 𝑅)} ≼ 1o
19	2, 18	eqbrtrdi 5146	. 2 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20		iffalse 4497	. . . 4 ⊢ (¬ 𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
21	20	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
22		safesnsupfidom1o.finite	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ Fin)
23		safesnsupfidom1o.small	. . . . 5 ⊢ (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
24		0elon 6387	. . . . . . . . 9 ⊢ ∅ ∈ On
25		eleq1 2816	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
26	24, 25	mpbiri 258	. . . . . . . 8 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
27		eleq1 2816	. . . . . . . . 9 ⊢ (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
28	4, 27	mpbiri 258	. . . . . . . 8 ⊢ (𝑂 = 1o → 𝑂 ∈ On)
29	26, 28	jaoi 857	. . . . . . 7 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
30		fidomtri 9946	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ 𝑂 ∈ On) → (𝐵 ≼ 𝑂 ↔ ¬ 𝑂 ≺ 𝐵))
31	29, 30	sylan2 593	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵 ≼ 𝑂 ↔ ¬ 𝑂 ≺ 𝐵))
32		breq2 5111	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ ∅))
33		domtr 8978	. . . . . . . . . 10 ⊢ ((𝐵 ≼ ∅ ∧ ∅ ≼ 1o) → 𝐵 ≼ 1o)
34	15, 33	mpan2 691	. . . . . . . . 9 ⊢ (𝐵 ≼ ∅ → 𝐵 ≼ 1o)
35	32, 34	biimtrdi 253	. . . . . . . 8 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
36		breq2 5111	. . . . . . . . 9 ⊢ (𝑂 = 1o → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ 1o))
37	36	biimpd 229	. . . . . . . 8 ⊢ (𝑂 = 1o → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
38	35, 37	jaoi 857	. . . . . . 7 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
39	38	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
40	31, 39	sylbird 260	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (¬ 𝑂 ≺ 𝐵 → 𝐵 ≼ 1o))
41	22, 23, 40	syl2anc 584	. . . 4 ⊢ (𝜑 → (¬ 𝑂 ≺ 𝐵 → 𝐵 ≼ 1o))
42	41	imp 406	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → 𝐵 ≼ 1o)
43	21, 42	eqbrtrd 5129	. 2 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
44	19, 43	pm2.61dan 812	1 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  ifcif 4488  {csn 4589   class class class wbr 5107  Oncon0 6332  1_oc1o 8427   ≈ cen 8915   ≼ cdom 8916   ≺ csdm 8917  Fincfn 8918  supcsup 9391

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892

This theorem is referenced by: (None)