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Theorem safesnsupfidom1o 43407
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 4537 . . . 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 9061 . . . . 5 (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4 1on 8517 . . . . . 6 1o ∈ On
5 domrefg 9026 . . . . . 6 (1o ∈ On → 1o ≼ 1o)
64, 5ax-mp 5 . . . . 5 1o ≼ 1o
7 endomtr 9051 . . . . 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 4722 . . . . . 6 (¬ sup(𝐵, 𝐴, 𝑅) ∈ V ↔ {sup(𝐵, 𝐴, 𝑅)} = ∅)
10 snex 5442 . . . . . . 7 {sup(𝐵, 𝐴, 𝑅)} ∈ V
11 eqeng 9025 . . . . . . 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 9139 . . . . . 6 (1o ∈ On → ∅ ≼ 1o)
154, 14ax-mp 5 . . . . 5 ∅ ≼ 1o
16 endomtr 9051 . . . . 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 5187 . 2 ((𝜑𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20 iffalse 4540 . . . 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 6440 . . . . . . . . 9 ∅ ∈ On
25 eleq1 2827 . . . . . . . . 9 (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
2624, 25mpbiri 258 . . . . . . . 8 (𝑂 = ∅ → 𝑂 ∈ On)
27 eleq1 2827 . . . . . . . . 9 (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
284, 27mpbiri 258 . . . . . . . 8 (𝑂 = 1o𝑂 ∈ On)
2926, 28jaoi 857 . . . . . . 7 ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
30 fidomtri 10031 . . . . . . 7 ((𝐵 ∈ Fin ∧ 𝑂 ∈ On) → (𝐵𝑂 ↔ ¬ 𝑂𝐵))
3129, 30sylan2 593 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵𝑂 ↔ ¬ 𝑂𝐵))
32 breq2 5152 . . . . . . . . 9 (𝑂 = ∅ → (𝐵𝑂𝐵 ≼ ∅))
33 domtr 9046 . . . . . . . . . 10 ((𝐵 ≼ ∅ ∧ ∅ ≼ 1o) → 𝐵 ≼ 1o)
3415, 33mpan2 691 . . . . . . . . 9 (𝐵 ≼ ∅ → 𝐵 ≼ 1o)
3532, 34biimtrdi 253 . . . . . . . 8 (𝑂 = ∅ → (𝐵𝑂𝐵 ≼ 1o))
36 breq2 5152 . . . . . . . . 9 (𝑂 = 1o → (𝐵𝑂𝐵 ≼ 1o))
3736biimpd 229 . . . . . . . 8 (𝑂 = 1o → (𝐵𝑂𝐵 ≼ 1o))
3835, 37jaoi 857 . . . . . . 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 5170 . 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 847   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  ifcif 4531  {csn 4631   class class class wbr 5148  Oncon0 6386  1oc1o 8498  cen 8981  cdom 8982  csdm 8983  Fincfn 8984  supcsup 9478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977
This theorem is referenced by: (None)
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