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Theorem suplocexpr 7526
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 3927 . . . . . 6 (𝑎 = 𝑤 → (𝑎 <Q 𝑢𝑤 <Q 𝑢))
54cbvrexv 2653 . . . . 5 (∃𝑎 (2nd𝐴)𝑎 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢)
65rabbii 2667 . . . 4 {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢} = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}
76opeq2i 3704 . . 3 (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
81, 2, 3, 7suplocexprlemex 7523 . 2 (𝜑 → ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ ∈ P)
91, 2, 3, 7suplocexprlemub 7524 . 2 (𝜑 → ∀𝑦𝐴 ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦)
101, 2, 3, 7suplocexprlemlub 7525 . . 3 (𝜑 → (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))
1110ralrimivw 2504 . 2 (𝜑 → ∀𝑦P (𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → ∃𝑧𝐴 𝑦<P 𝑧))
12 breq1 3927 . . . . . 6 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
1312notbid 656 . . . . 5 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (¬ 𝑥<P 𝑦 ↔ ¬ ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩<P 𝑦))
1413ralbidv 2435 . . . 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 3928 . . . . . 6 (𝑥 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩ → (𝑦<P 𝑥𝑦<P (1st𝐴), {𝑢Q ∣ ∃𝑎 (2nd𝐴)𝑎 <Q 𝑢}⟩))
1615imbi1d 230 . . . . 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 2435 . . . 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 464 . . 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 2784 . 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 1214 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 103  wo 697   = wceq 1331  wex 1468  wcel 1480  wral 2414  wrex 2415  {crab 2418  cop 3525   cuni 3731   cint 3766   class class class wbr 3924  cima 4537  1st c1st 6029  2nd c2nd 6030  Qcnq 7081   <Q cltq 7086  Pcnp 7092  <P cltp 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iltp 7271
This theorem is referenced by:  suplocsrlempr  7608
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