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Theorem safesnsupfiub 41762
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
safesnsupfiub.small (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
safesnsupfiub.finite (𝜑𝐵 ∈ Fin)
safesnsupfiub.subset (𝜑𝐵𝐴)
safesnsupfiub.ordered (𝜑𝑅 Or 𝐴)
safesnsupfiub.ub (𝜑 → ∀𝑥𝐵𝑦𝐶 𝑥𝑅𝑦)
Assertion
Ref Expression
safesnsupfiub (𝜑 → ∀𝑥 ∈ if (𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦𝐶 𝑥𝑅𝑦)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem safesnsupfiub
StepHypRef Expression
1 safesnsupfiub.ub . 2 (𝜑 → ∀𝑥𝐵𝑦𝐶 𝑥𝑅𝑦)
2 safesnsupfiub.small . . . . . 6 (𝜑 → (𝑂 = ∅ ∨ 𝑂 = 1o))
3 safesnsupfiub.finite . . . . . 6 (𝜑𝐵 ∈ Fin)
4 safesnsupfiub.subset . . . . . 6 (𝜑𝐵𝐴)
5 safesnsupfiub.ordered . . . . . 6 (𝜑𝑅 Or 𝐴)
62, 3, 4, 5safesnsupfiss 41761 . . . . 5 (𝜑 → if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) ⊆ 𝐵)
76sseld 3948 . . . 4 (𝜑 → (𝑥 ∈ if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) → 𝑥𝐵))
87imim1d 82 . . 3 (𝜑 → ((𝑥𝐵 → ∀𝑦𝐶 𝑥𝑅𝑦) → (𝑥 ∈ if(𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵) → ∀𝑦𝐶 𝑥𝑅𝑦)))
98ralimdv2 3161 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐶 𝑥𝑅𝑦 → ∀𝑥 ∈ if (𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦𝐶 𝑥𝑅𝑦))
101, 9mpd 15 1 (𝜑 → ∀𝑥 ∈ if (𝑂𝐵, {sup(𝐵, 𝐴, 𝑅)}, 𝐵)∀𝑦𝐶 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1542  wcel 2107  wral 3065  wss 3915  c0 4287  ifcif 4491  {csn 4591   class class class wbr 5110   Or wor 5549  1oc1o 8410  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-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  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-br 5111  df-opab 5173  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-riota 7318  df-om 7808  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385
This theorem is referenced by: (None)
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