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Theorem safesnsupfidom1o 41763
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 4497 . . . 4 (𝑂𝐵 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
21adantl 483 . . 3 ((𝜑𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = {sup(𝐵, 𝐴, 𝑅)})
3 ensn1g 8970 . . . . 5 (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ 1o)
4 1on 8429 . . . . . 6 1o ∈ On
5 domrefg 8934 . . . . . 6 (1o ∈ On → 1o ≼ 1o)
64, 5ax-mp 5 . . . . 5 1o ≼ 1o
7 endomtr 8959 . . . . 5 (({sup(𝐵, 𝐴, 𝑅)} ≈ 1o ∧ 1o ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
83, 6, 7sylancl 587 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
9 snprc 4683 . . . . . 6 (¬ sup(𝐵, 𝐴, 𝑅) ∈ V ↔ {sup(𝐵, 𝐴, 𝑅)} = ∅)
10 snex 5393 . . . . . . 7 {sup(𝐵, 𝐴, 𝑅)} ∈ V
11 eqeng 8933 . . . . . . 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 216 . . . . 5 (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≈ ∅)
14 0domg 9051 . . . . . 6 (1o ∈ On → ∅ ≼ 1o)
154, 14ax-mp 5 . . . . 5 ∅ ≼ 1o
16 endomtr 8959 . . . . 5 (({sup(𝐵, 𝐴, 𝑅)} ≈ ∅ ∧ ∅ ≼ 1o) → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
1713, 15, 16sylancl 587 . . . 4 (¬ sup(𝐵, 𝐴, 𝑅) ∈ V → {sup(𝐵, 𝐴, 𝑅)} ≼ 1o)
188, 17pm2.61i 182 . . 3 {sup(𝐵, 𝐴, 𝑅)} ≼ 1o
192, 18eqbrtrdi 5149 . 2 ((𝜑𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
20 iffalse 4500 . . . 4 𝑂𝐵 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) = 𝐵)
2120adantl 483 . . 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 6376 . . . . . . . . 9 ∅ ∈ On
25 eleq1 2826 . . . . . . . . 9 (𝑂 = ∅ → (𝑂 ∈ On ↔ ∅ ∈ On))
2624, 25mpbiri 258 . . . . . . . 8 (𝑂 = ∅ → 𝑂 ∈ On)
27 eleq1 2826 . . . . . . . . 9 (𝑂 = 1o → (𝑂 ∈ On ↔ 1o ∈ On))
284, 27mpbiri 258 . . . . . . . 8 (𝑂 = 1o𝑂 ∈ On)
2926, 28jaoi 856 . . . . . . 7 ((𝑂 = ∅ ∨ 𝑂 = 1o) → 𝑂 ∈ On)
30 fidomtri 9936 . . . . . . 7 ((𝐵 ∈ Fin ∧ 𝑂 ∈ On) → (𝐵𝑂 ↔ ¬ 𝑂𝐵))
3129, 30sylan2 594 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵𝑂 ↔ ¬ 𝑂𝐵))
32 breq2 5114 . . . . . . . . 9 (𝑂 = ∅ → (𝐵𝑂𝐵 ≼ ∅))
33 domtr 8954 . . . . . . . . . 10 ((𝐵 ≼ ∅ ∧ ∅ ≼ 1o) → 𝐵 ≼ 1o)
3415, 33mpan2 690 . . . . . . . . 9 (𝐵 ≼ ∅ → 𝐵 ≼ 1o)
3532, 34syl6bi 253 . . . . . . . 8 (𝑂 = ∅ → (𝐵𝑂𝐵 ≼ 1o))
36 breq2 5114 . . . . . . . . 9 (𝑂 = 1o → (𝐵𝑂𝐵 ≼ 1o))
3736biimpd 228 . . . . . . . 8 (𝑂 = 1o → (𝐵𝑂𝐵 ≼ 1o))
3835, 37jaoi 856 . . . . . . 7 ((𝑂 = ∅ ∨ 𝑂 = 1o) → (𝐵𝑂𝐵 ≼ 1o))
3938adantl 483 . . . . . 6 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (𝐵𝑂𝐵 ≼ 1o))
4031, 39sylbird 260 . . . . 5 ((𝐵 ∈ Fin ∧ (𝑂 = ∅ ∨ 𝑂 = 1o)) → (¬ 𝑂𝐵𝐵 ≼ 1o))
4122, 23, 40syl2anc 585 . . . 4 (𝜑 → (¬ 𝑂𝐵𝐵 ≼ 1o))
4241imp 408 . . 3 ((𝜑 ∧ ¬ 𝑂𝐵) → 𝐵 ≼ 1o)
4321, 42eqbrtrd 5132 . 2 ((𝜑 ∧ ¬ 𝑂𝐵) → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
4419, 43pm2.61dan 812 1 (𝜑 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ≼ 1o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  Vcvv 3448  c0 4287  ifcif 4491  {csn 4591   class class class wbr 5110  Oncon0 6322  1oc1o 8410  cen 8887  cdom 8888  csdm 8889  Fincfn 8890  supcsup 9383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882
This theorem is referenced by: (None)
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