Theorem safesnsupfidom1o 43658

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 4485	. . . 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 8959	. . . . 5 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4		1on 8409	. . . . . 6 ⊢ 1o ∈ On
5		domrefg 8924	. . . . . 6 ⊢ (1o ∈ On → 1o ≼ 1o)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1o ≼ 1o
7		endomtr 8949	. . . . 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 4674	. . . . . 6 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V ↔ {sup(𝐵, 𝐴, 𝑅)} = ∅)
10		snex 5381	. . . . . . 7 ⊢ {sup(𝐵, 𝐴, 𝑅)} ∈ V
11		eqeng 8923	. . . . . . 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 9032	. . . . . 6 ⊢ (1o ∈ On → ∅ ≼ 1o)
15	4, 14	ax-mp 5	. . . . 5 ⊢ ∅ ≼ 1o
16		endomtr 8949	. . . . 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 5137	. 2 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20		iffalse 4488	. . . 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 6372	. . . . . . . . 9 ⊢ ∅ ∈ On
25		eleq1 2824	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
26	24, 25	mpbiri 258	. . . . . . . 8 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
27		eleq1 2824	. . . . . . . . 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 9905	. . . . . . 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 5102	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ ∅))
33		domtr 8944	. . . . . . . . . 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 5102	. . . . . . . . 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 5120	. 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 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ifcif 4479  {csn 4580   class class class wbr 5098  Oncon0 6317  1_oc1o 8390   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  Fincfn 8883  supcsup 9343

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851

This theorem is referenced by: (None)