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Theorem suplocexpr 7723
Description: An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocexpr.ub (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
suplocexpr.loc (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
Assertion
Ref Expression
suplocexpr (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
Distinct variable groups:   𝑦,𝐴,𝑧,𝑥   𝜑,𝑦,𝑧,𝑥

Proof of Theorem suplocexpr
Dummy variables 𝑎 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.m . . 3 (𝜑 → ∃𝑥 𝑥𝐴)
2 suplocexpr.ub . . 3 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
3 suplocexpr.loc . . 3 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
4 breq1 4006 . . . . . 6 (𝑎 = 𝑤 → (𝑎 <Q 𝑢𝑤 <Q 𝑢))
54cbvrexv 2704 . . . . 5 (∃𝑎 (2nd𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢)
65rabbii 2723 . . . 4 {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢} = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}
76opeq2i 3782 . . 3 (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
81, 2, 3, 7suplocexprlemex 7720 . 2 (𝜑 → ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
91, 2, 3, 7suplocexprlemub 7721 . 2 (𝜑 → ∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
101, 2, 3, 7suplocexprlemlub 7722 . . 3 (𝜑 → (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))
1110ralrimivw 2551 . 2 (𝜑 → ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))
12 breq1 4006 . . . . . 6 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
1312notbid 667 . . . . 5 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (¬ 𝑥<P 𝑦 ↔ ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
1413ralbidv 2477 . . . 4 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ↔ ∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
15 breq2 4007 . . . . . 6 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (𝑦<P 𝑥𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩))
1615imbi1d 231 . . . . 5 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ((𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧) ↔ (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧)))
1716ralbidv 2477 . . . 4 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧) ↔ ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧)))
1814, 17anbi12d 473 . . 3 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ((∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)) ↔ (∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦 ∧ ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))))
1918rspcev 2841 . 2 ((⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ ∈ P ∧ (∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦 ∧ ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))) → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
208, 9, 11, 19syl12anc 1236 1 (𝜑 → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  {crab 2459  cop 3595   cuni 3809   cint 3844   class class class wbr 4003  cima 4629  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   <Q cltq 7283  Pcnp 7289  <P cltp 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iltp 7468
This theorem is referenced by:  suplocsrlempr  7805
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