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Theorem suplocexprlemub 7543
Description: Lemma for suplocexpr 7545. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-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
suplocexprlemub (𝜑 → ∀𝑦𝐴 ¬ 𝐵<P 𝑦)
Distinct variable groups:   𝑢,𝐴,𝑤,𝑦   𝑥,𝐴,𝑧,𝑢,𝑦   𝑤,𝐵   𝜑,𝑢,𝑤,𝑦   𝜑,𝑥,𝑧   𝑧,𝑤
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem suplocexprlemub
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → 𝐵<P 𝑦)
2 suplocexpr.m . . . . . . . 8 (𝜑 → ∃𝑥 𝑥𝐴)
3 suplocexpr.ub . . . . . . . 8 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
4 suplocexpr.loc . . . . . . . 8 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
5 suplocexpr.b . . . . . . . 8 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
62, 3, 4, 5suplocexprlemex 7542 . . . . . . 7 (𝜑𝐵P)
76ad2antrr 479 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → 𝐵P)
82, 3, 4suplocexprlemss 7535 . . . . . . . 8 (𝜑𝐴P)
98ad2antrr 479 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → 𝐴P)
10 simplr 519 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → 𝑦𝐴)
119, 10sseldd 3098 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → 𝑦P)
12 ltdfpr 7326 . . . . . 6 ((𝐵P𝑦P) → (𝐵<P 𝑦 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦))))
137, 11, 12syl2anc 408 . . . . 5 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → (𝐵<P 𝑦 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦))))
141, 13mpbid 146 . . . 4 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))
15 simprrl 528 . . . . . . . 8 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → 𝑠 ∈ (2nd𝐵))
165suplocexprlem2b 7534 . . . . . . . . . . 11 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
178, 16syl 14 . . . . . . . . . 10 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
1817eleq2d 2209 . . . . . . . . 9 (𝜑 → (𝑠 ∈ (2nd𝐵) ↔ 𝑠 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
1918ad3antrrr 483 . . . . . . . 8 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → (𝑠 ∈ (2nd𝐵) ↔ 𝑠 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
2015, 19mpbid 146 . . . . . . 7 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → 𝑠 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
21 breq2 3933 . . . . . . . . 9 (𝑢 = 𝑠 → (𝑤 <Q 𝑢𝑤 <Q 𝑠))
2221rexbidv 2438 . . . . . . . 8 (𝑢 = 𝑠 → (∃𝑤 (2nd𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑠))
2322elrab 2840 . . . . . . 7 (𝑠 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ↔ (𝑠Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑠))
2420, 23sylib 121 . . . . . 6 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → (𝑠Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑠))
2524simprd 113 . . . . 5 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → ∃𝑤 (2nd𝐴)𝑤 <Q 𝑠)
26 simprrr 529 . . . . . . . 8 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → 𝑠 ∈ (1st𝑦))
2726adantr 274 . . . . . . 7 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑠 ∈ (1st𝑦))
28 simprr 521 . . . . . . . 8 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑤 <Q 𝑠)
2911ad2antrr 479 . . . . . . . . . 10 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑦P)
30 prop 7295 . . . . . . . . . 10 (𝑦P → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ P)
3129, 30syl 14 . . . . . . . . 9 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ P)
32 eleq2 2203 . . . . . . . . . 10 (𝑡 = (2nd𝑦) → (𝑤𝑡𝑤 ∈ (2nd𝑦)))
33 simprl 520 . . . . . . . . . . 11 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑤 (2nd𝐴))
34 vex 2689 . . . . . . . . . . . 12 𝑤 ∈ V
3534elint2 3778 . . . . . . . . . . 11 (𝑤 (2nd𝐴) ↔ ∀𝑡 ∈ (2nd𝐴)𝑤𝑡)
3633, 35sylib 121 . . . . . . . . . 10 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → ∀𝑡 ∈ (2nd𝐴)𝑤𝑡)
37 fo2nd 6056 . . . . . . . . . . . . 13 2nd :V–onto→V
38 fofun 5346 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
3937, 38ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
40 vex 2689 . . . . . . . . . . . . 13 𝑦 ∈ V
41 fof 5345 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → 2nd :V⟶V)
4237, 41ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟶V
4342fdmi 5280 . . . . . . . . . . . . 13 dom 2nd = V
4440, 43eleqtrri 2215 . . . . . . . . . . . 12 𝑦 ∈ dom 2nd
45 funfvima 5649 . . . . . . . . . . . 12 ((Fun 2nd𝑦 ∈ dom 2nd ) → (𝑦𝐴 → (2nd𝑦) ∈ (2nd𝐴)))
4639, 44, 45mp2an 422 . . . . . . . . . . 11 (𝑦𝐴 → (2nd𝑦) ∈ (2nd𝐴))
4746ad4antlr 486 . . . . . . . . . 10 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → (2nd𝑦) ∈ (2nd𝐴))
4832, 36, 47rspcdva 2794 . . . . . . . . 9 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑤 ∈ (2nd𝑦))
49 prcunqu 7305 . . . . . . . . 9 ((⟨(1st𝑦), (2nd𝑦)⟩ ∈ P𝑤 ∈ (2nd𝑦)) → (𝑤 <Q 𝑠𝑠 ∈ (2nd𝑦)))
5031, 48, 49syl2anc 408 . . . . . . . 8 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → (𝑤 <Q 𝑠𝑠 ∈ (2nd𝑦)))
5128, 50mpd 13 . . . . . . 7 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑠 ∈ (2nd𝑦))
5227, 51jca 304 . . . . . 6 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → (𝑠 ∈ (1st𝑦) ∧ 𝑠 ∈ (2nd𝑦)))
53 simplrl 524 . . . . . . 7 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑠Q)
54 prdisj 7312 . . . . . . 7 ((⟨(1st𝑦), (2nd𝑦)⟩ ∈ P𝑠Q) → ¬ (𝑠 ∈ (1st𝑦) ∧ 𝑠 ∈ (2nd𝑦)))
5531, 53, 54syl2anc 408 . . . . . 6 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → ¬ (𝑠 ∈ (1st𝑦) ∧ 𝑠 ∈ (2nd𝑦)))
5652, 55pm2.21fal 1351 . . . . 5 (((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) ∧ (𝑤 (2nd𝐴) ∧ 𝑤 <Q 𝑠)) → ⊥)
5725, 56rexlimddv 2554 . . . 4 ((((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝑦)))) → ⊥)
5814, 57rexlimddv 2554 . . 3 (((𝜑𝑦𝐴) ∧ 𝐵<P 𝑦) → ⊥)
5958inegd 1350 . 2 ((𝜑𝑦𝐴) → ¬ 𝐵<P 𝑦)
6059ralrimiva 2505 1 (𝜑 → ∀𝑦𝐴 ¬ 𝐵<P 𝑦)
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-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  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-nul 3364  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-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  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-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iltp 7290
This theorem is referenced by:  suplocexpr  7545
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