Theorem safesnsupfidom1o 42770

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 4530	. . . 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 9035	. . . . 5 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4		1on 8492	. . . . . 6 ⊢ 1o ∈ On
5		domrefg 8999	. . . . . 6 ⊢ (1o ∈ On → 1o ≼ 1o)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1o ≼ 1o
7		endomtr 9024	. . . . 5 ⊢ (({sup(𝐵, 𝐴, 𝑅)} ≈ 1o ∧ 1o ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
8	3, 6, 7	sylancl 585	. . . 4 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
9		snprc 4717	. . . . . 6 ⊢ (¬ sup(𝐵, 𝐴, 𝑅) ∈ V ↔ {sup(𝐵, 𝐴, 𝑅)} = ∅)
10		snex 5427	. . . . . . 7 ⊢ {sup(𝐵, 𝐴, 𝑅)} ∈ V
11		eqeng 8998	. . . . . . 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 9116	. . . . . 6 ⊢ (1o ∈ On → ∅ ≼ 1o)
15	4, 14	ax-mp 5	. . . . 5 ⊢ ∅ ≼ 1o
16		endomtr 9024	. . . . 5 ⊢ (({sup(𝐵, 𝐴, 𝑅)} ≈ ∅ ∧ ∅ ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
17	13, 15, 16	sylancl 585	. . . 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 5181	. 2 ⊢ ((𝜑 ∧ 𝑂 ≺ 𝐵) → if(𝑂 ≺ 𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20		iffalse 4533	. . . 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 6417	. . . . . . . . 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 856	. . . . . . 7 ⊢ ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
30		fidomtri 10008	. . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ 𝑂 ∈ On) → (𝐵 ≼ 𝑂 ↔ ¬ 𝑂 ≺ 𝐵))
31	29, 30	sylan2 592	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵 ≼ 𝑂 ↔ ¬ 𝑂 ≺ 𝐵))
32		breq2 5146	. . . . . . . . 9 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ ∅))
33		domtr 9019	. . . . . . . . . 10 ⊢ ((𝐵 ≼ ∅ ∧ ∅ ≼ 1o) → 𝐵 ≼ 1o)
34	15, 33	mpan2 690	. . . . . . . . 9 ⊢ (𝐵 ≼ ∅ → 𝐵 ≼ 1o)
35	32, 34	syl6bi 253	. . . . . . . 8 ⊢ (𝑂 = ∅ → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
36		breq2 5146	. . . . . . . . 9 ⊢ (𝑂 = 1o → (𝐵 ≼ 𝑂 ↔ 𝐵 ≼ 1o))
37	36	biimpd 228	. . . . . . . 8 ⊢ (𝑂 = 1o → (𝐵 ≼ 𝑂 → 𝐵 ≼ 1o))
38	35, 37	jaoi 856	. . . . . . 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 583	. . . 4 ⊢ (𝜑 → (¬ 𝑂 ≺ 𝐵 → 𝐵 ≼ 1o))
42	41	imp 406	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑂 ≺ 𝐵) → 𝐵 ≼ 1o)
43	21, 42	eqbrtrd 5164	. 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 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  Vcvv 3469  ∅c0 4318  ifcif 4524  {csn 4624   class class class wbr 5142  Oncon0 6363  1_oc1o 8473   ≈ cen 8952   ≼ cdom 8953   ≺ csdm 8954  Fincfn 8955  supcsup 9455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7865  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9954

This theorem is referenced by: (None)