Theorem safesnsupfidom1o 42912

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 4535	. . . 4 ⊢ (𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
2	1	adantl 480	. . 3 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
3		ensn1g 9042	. . . . 5 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4		1on 8497	. . . . . 6 ⊢ 1o ∈ On
5		domrefg 9006	. . . . . 6 ⊢ (1o ∈ On → 1o ≼ 1o)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1o ≼ 1o
7		endomtr 9031	. . . . 5 ⊢ (({sup(𝐵, 𝐴, 𝑅)} ≈ 1o ∧ 1o ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
8	3, 6, 7	sylancl 584	. . . 4 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
9		snprc 4722	. . . . . 6 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V ↔ {sup(𝐵, 𝐴, 𝑅)} = ∅)
10		snex 5432	. . . . . . 7 ⊢ {sup(𝐵, 𝐴, 𝑅)} ∈ V
11		eqeng 9005	. . . . . . 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 216	. . . . 5 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅)
14		0domg 9123	. . . . . 6 ⊢ (1o ∈ On → ∅ ≼ 1o)
15	4, 14	ax-mp 5	. . . . 5 ⊢ ∅ ≼ 1o
16		endomtr 9031	. . . . 5 ⊢ (({sup(𝐵, 𝐴, 𝑅)} ≈ ∅ ∧ ∅ ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
17	13, 15, 16	sylancl 584	. . . 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 5187	. 2 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20		iffalse 4538	. . . 4 ⊢ (¬ 𝑂 ≺ 𝐵 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
21	20	adantl 480	. . 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 6423	. . . . . . . . 9 ⊢ ∅ ∈ On
25		eleq1 2813	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
26	24, 25	mpbiri 257	. . . . . . . 8 ⊢ (𝑂 = ∅ → 𝑂 ∈ On)
27		eleq1 2813	. . . . . . . . 9 ⊢ (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
28	4, 27	mpbiri 257	. . . . . . . 8 ⊢ (𝑂 = 1o → 𝑂 ∈ On)
29	26, 28	jaoi 855	. . . . . . 7 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
30		fidomtri 10016	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ 𝑂 ∈ On) → (𝐵 ≼ 𝑂 ↔ ¬ 𝑂 ≺ 𝐵))
31	29, 30	sylan2 591	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵 ≼ 𝑂 ↔ ¬ 𝑂 ≺ 𝐵))
32		breq2 5152	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ ∅))
33		domtr 9026	. . . . . . . . . 10 ⊢ ((𝐵 ≼ ∅ ∧ ∅ ≼ 1o) → 𝐵 ≼ 1o)
34	15, 33	mpan2 689	. . . . . . . . 9 ⊢ (𝐵 ≼ ∅ → 𝐵 ≼ 1o)
35	32, 34	biimtrdi 252	. . . . . . . 8 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
36		breq2 5152	. . . . . . . . 9 ⊢ (𝑂 = 1o → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ 1o))
37	36	biimpd 228	. . . . . . . 8 ⊢ (𝑂 = 1o → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
38	35, 37	jaoi 855	. . . . . . 7 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
39	38	adantl 480	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
40	31, 39	sylbird 259	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (¬ 𝑂 ≺ 𝐵 → 𝐵 ≼ 1o))
41	22, 23, 40	syl2anc 582	. . . 4 ⊢ (𝜑 → (¬ 𝑂 ≺ 𝐵 → 𝐵 ≼ 1o))
42	41	imp 405	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → 𝐵 ≼ 1o)
43	21, 42	eqbrtrd 5170	. 2 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
44	19, 43	pm2.61dan 811	1 ⊢ (𝜑 → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  Vcvv 3463  ∅c0 4323  ifcif 4529  {csn 4629   class class class wbr 5148  Oncon0 6369  1_oc1o 8478   ≈ cen 8959   ≼ cdom 8960   ≺ csdm 8961  Fincfn 8962  supcsup 9463

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 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739

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-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-om 7870  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-card 9962

This theorem is referenced by: (None)