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Theorem safesnsupfidom1o 43430
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 (𝜑𝐵 ∈ Fin)
Assertion
Ref Expression
safesnsupfidom1o (𝜑 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)

Proof of Theorem safesnsupfidom1o
StepHypRef Expression
1 iftrue 4531 . . . 4 (𝑂𝐵 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
21adantl 481 . . 3 ((𝜑𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
3 ensn1g 9062 . . . . 5 (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4 1on 8518 . . . . . 6 1o ∈ On
5 domrefg 9027 . . . . . 6 (1o ∈ On → 1o ≼ 1o)
64, 5ax-mp 5 . . . . 5 1o ≼ 1o
7 endomtr 9052 . . . . 5 (({sup(𝐵, 𝐴, 𝑅)} ≈ 1o ∧ 1o ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
83, 6, 7sylancl 586 . . . 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 5436 . . . . . . 7 {sup(𝐵, 𝐴, 𝑅)} ∈ V
11 eqeng 9026 . . . . . . 7 ({sup(𝐵, 𝐴, 𝑅)} ∈ V → ({sup(𝐵, 𝐴, 𝑅)} = ∅ → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅))
1210, 11ax-mp 5 . . . . . 6 ({sup(𝐵, 𝐴, 𝑅)} = ∅ → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅)
139, 12sylbi 217 . . . . 5 (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅)
14 0domg 9140 . . . . . 6 (1o ∈ On → ∅ ≼ 1o)
154, 14ax-mp 5 . . . . 5 ∅ ≼ 1o
16 endomtr 9052 . . . . 5 (({sup(𝐵, 𝐴, 𝑅)} ≈ ∅ ∧ ∅ ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
1713, 15, 16sylancl 586 . . . 4 (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
188, 17pm2.61i 182 . . 3 {sup(𝐵, 𝐴, 𝑅)} ≼ 1o
192, 18eqbrtrdi 5182 . 2 ((𝜑𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20 iffalse 4534 . . . 4 𝑂𝐵 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
2120adantl 481 . . 3 ((𝜑 ∧ ¬ 𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
22 safesnsupfidom1o.finite . . . . 5 (𝜑𝐵 ∈ Fin)
23 safesnsupfidom1o.small . . . . 5 (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
24 0elon 6438 . . . . . . . . 9 ∅ ∈ On
25 eleq1 2829 . . . . . . . . 9 (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
2624, 25mpbiri 258 . . . . . . . 8 (𝑂 = ∅ → 𝑂 ∈ On)
27 eleq1 2829 . . . . . . . . 9 (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
284, 27mpbiri 258 . . . . . . . 8 (𝑂 = 1o𝑂 ∈ On)
2926, 28jaoi 858 . . . . . . 7 ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
30 fidomtri 10033 . . . . . . 7 ((𝐵 ∈ Fin ∧ 𝑂 ∈ On) → (𝐵𝑂 ↔ ¬ 𝑂𝐵))
3129, 30sylan2 593 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵𝑂 ↔ ¬ 𝑂𝐵))
32 breq2 5147 . . . . . . . . 9 (𝑂 = ∅ → (𝐵𝑂𝐵 ≼ ∅))
33 domtr 9047 . . . . . . . . . 10 ((𝐵 ≼ ∅ ∧ ∅ ≼ 1o) → 𝐵 ≼ 1o)
3415, 33mpan2 691 . . . . . . . . 9 (𝐵 ≼ ∅ → 𝐵 ≼ 1o)
3532, 34biimtrdi 253 . . . . . . . 8 (𝑂 = ∅ → (𝐵𝑂𝐵 ≼ 1o))
36 breq2 5147 . . . . . . . . 9 (𝑂 = 1o → (𝐵𝑂𝐵 ≼ 1o))
3736biimpd 229 . . . . . . . 8 (𝑂 = 1o → (𝐵𝑂𝐵 ≼ 1o))
3835, 37jaoi 858 . . . . . . 7 ((𝑂 = ∅ ∨ 𝑂 = 1o) → (𝐵𝑂𝐵 ≼ 1o))
3938adantl 481 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵𝑂𝐵 ≼ 1o))
4031, 39sylbird 260 . . . . 5 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (¬ 𝑂𝐵𝐵 ≼ 1o))
4122, 23, 40syl2anc 584 . . . 4 (𝜑 → (¬ 𝑂𝐵𝐵 ≼ 1o))
4241imp 406 . . 3 ((𝜑 ∧ ¬ 𝑂𝐵) → 𝐵 ≼ 1o)
4321, 42eqbrtrd 5165 . 2 ((𝜑 ∧ ¬ 𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
4419, 43pm2.61dan 813 1 (𝜑 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  ifcif 4525  {csn 4626   class class class wbr 5143  Oncon0 6384  1oc1o 8499  cen 8982  cdom 8983  csdm 8984  Fincfn 8985  supcsup 9480
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979
This theorem is referenced by: (None)
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