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Theorem suplocexprlemdisj 7540
Description: Lemma for suplocexpr 7545. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-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)))
suplocexpr.b 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
Assertion
Ref Expression
suplocexprlemdisj (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))
Distinct variable groups:   𝑤,𝐴,𝑢   𝑥,𝐴,𝑦   𝑤,𝐵   𝜑,𝑞,𝑤   𝜑,𝑥,𝑦   𝑢,𝑞
Allowed substitution hints:   𝜑(𝑧,𝑢)   𝐴(𝑧,𝑞)   𝐵(𝑥,𝑦,𝑧,𝑢,𝑞)

Proof of Theorem suplocexprlemdisj
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 520 . . . . 5 (((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) → 𝑞 (1st𝐴))
2 suplocexprlemell 7533 . . . . 5 (𝑞 (1st𝐴) ↔ ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
31, 2sylib 121 . . . 4 (((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) → ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
4 simprr 521 . . . . . 6 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑞 ∈ (1st𝑠))
5 simplrr 525 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑞 ∈ (2nd𝐵))
6 suplocexpr.m . . . . . . . . . . . . 13 (𝜑 → ∃𝑥 𝑥𝐴)
7 suplocexpr.ub . . . . . . . . . . . . 13 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
8 suplocexpr.loc . . . . . . . . . . . . 13 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
96, 7, 8suplocexprlemss 7535 . . . . . . . . . . . 12 (𝜑𝐴P)
109ad3antrrr 483 . . . . . . . . . . 11 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝐴P)
11 suplocexpr.b . . . . . . . . . . . . 13 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
1211suplocexprlem2b 7534 . . . . . . . . . . . 12 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
1312eleq2d 2209 . . . . . . . . . . 11 (𝐴P → (𝑞 ∈ (2nd𝐵) ↔ 𝑞 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
1410, 13syl 14 . . . . . . . . . 10 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → (𝑞 ∈ (2nd𝐵) ↔ 𝑞 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
15 breq2 3933 . . . . . . . . . . . 12 (𝑢 = 𝑞 → (𝑤 <Q 𝑢𝑤 <Q 𝑞))
1615rexbidv 2438 . . . . . . . . . . 11 (𝑢 = 𝑞 → (∃𝑤 (2nd𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑞))
1716elrab 2840 . . . . . . . . . 10 (𝑞 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ↔ (𝑞Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑞))
1814, 17syl6bb 195 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → (𝑞 ∈ (2nd𝐵) ↔ (𝑞Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑞)))
195, 18mpbid 146 . . . . . . . 8 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → (𝑞Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑞))
2019simprd 113 . . . . . . 7 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ∃𝑤 (2nd𝐴)𝑤 <Q 𝑞)
21 simprr 521 . . . . . . . 8 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑤 <Q 𝑞)
2210adantr 274 . . . . . . . . . . 11 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝐴P)
23 simplrl 524 . . . . . . . . . . 11 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑠𝐴)
2422, 23sseldd 3098 . . . . . . . . . 10 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑠P)
25 prop 7295 . . . . . . . . . 10 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
2624, 25syl 14 . . . . . . . . 9 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
27 eleq2 2203 . . . . . . . . . 10 (𝑡 = (2nd𝑠) → (𝑤𝑡𝑤 ∈ (2nd𝑠)))
28 simprl 520 . . . . . . . . . . 11 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑤 (2nd𝐴))
29 vex 2689 . . . . . . . . . . . 12 𝑤 ∈ V
3029elint2 3778 . . . . . . . . . . 11 (𝑤 (2nd𝐴) ↔ ∀𝑡 ∈ (2nd𝐴)𝑤𝑡)
3128, 30sylib 121 . . . . . . . . . 10 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → ∀𝑡 ∈ (2nd𝐴)𝑤𝑡)
32 fo2nd 6056 . . . . . . . . . . . . . 14 2nd :V–onto→V
33 fofun 5346 . . . . . . . . . . . . . 14 (2nd :V–onto→V → Fun 2nd )
3432, 33ax-mp 5 . . . . . . . . . . . . 13 Fun 2nd
35 vex 2689 . . . . . . . . . . . . . 14 𝑠 ∈ V
36 fof 5345 . . . . . . . . . . . . . . . 16 (2nd :V–onto→V → 2nd :V⟶V)
3732, 36ax-mp 5 . . . . . . . . . . . . . . 15 2nd :V⟶V
3837fdmi 5280 . . . . . . . . . . . . . 14 dom 2nd = V
3935, 38eleqtrri 2215 . . . . . . . . . . . . 13 𝑠 ∈ dom 2nd
40 funfvima 5649 . . . . . . . . . . . . 13 ((Fun 2nd𝑠 ∈ dom 2nd ) → (𝑠𝐴 → (2nd𝑠) ∈ (2nd𝐴)))
4134, 39, 40mp2an 422 . . . . . . . . . . . 12 (𝑠𝐴 → (2nd𝑠) ∈ (2nd𝐴))
4241ad2antrl 481 . . . . . . . . . . 11 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → (2nd𝑠) ∈ (2nd𝐴))
4342adantr 274 . . . . . . . . . 10 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → (2nd𝑠) ∈ (2nd𝐴))
4427, 31, 43rspcdva 2794 . . . . . . . . 9 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑤 ∈ (2nd𝑠))
45 prcunqu 7305 . . . . . . . . 9 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑤 ∈ (2nd𝑠)) → (𝑤 <Q 𝑞𝑞 ∈ (2nd𝑠)))
4626, 44, 45syl2anc 408 . . . . . . . 8 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → (𝑤 <Q 𝑞𝑞 ∈ (2nd𝑠)))
4721, 46mpd 13 . . . . . . 7 (((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑞 ∈ (2nd𝑠))
4820, 47rexlimddv 2554 . . . . . 6 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑞 ∈ (2nd𝑠))
494, 48jca 304 . . . . 5 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
50 simprl 520 . . . . . . . 8 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑠𝐴)
5110, 50sseldd 3098 . . . . . . 7 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑠P)
5251, 25syl 14 . . . . . 6 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
53 simpllr 523 . . . . . 6 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑞Q)
54 prdisj 7312 . . . . . 6 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
5552, 53, 54syl2anc 408 . . . . 5 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
5649, 55pm2.21fal 1351 . . . 4 ((((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ⊥)
573, 56rexlimddv 2554 . . 3 (((𝜑𝑞Q) ∧ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵))) → ⊥)
5857inegd 1350 . 2 ((𝜑𝑞Q) → ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))
5958ralrimiva 2505 1 (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697   = wceq 1331  wfal 1336  wex 1468  wcel 1480  wral 2416  wrex 2417  {crab 2420  Vcvv 2686  wss 3071  cop 3530   cuni 3736   cint 3771   class class class wbr 3929  dom cdm 4539  cima 4542  Fun wfun 5117  wf 5119  ontowfo 5121  cfv 5123  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   <Q cltq 7105  Pcnp 7111  <P cltp 7115
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 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7124  df-nqqs 7168  df-ltnqqs 7173  df-inp 7286  df-iltp 7290
This theorem is referenced by:  suplocexprlemex  7542
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