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Theorem recexprlemopl 7621
Description: The lower cut of 𝐵 is open. Lemma for recexpr 7634. (Contributed by Jim Kingdon, 28-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemopl ((𝐴P𝑞Q𝑞 ∈ (1st𝐵)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))
Distinct variable groups:   𝑟,𝑞,𝑥,𝑦,𝐴   𝐵,𝑞,𝑟,𝑥,𝑦

Proof of Theorem recexprlemopl
StepHypRef Expression
1 recexpr.1 . . . 4 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
21recexprlemell 7618 . . 3 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
3 ltbtwnnqq 7411 . . . . . 6 (𝑞 <Q 𝑦 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 <Q 𝑦))
43biimpi 120 . . . . 5 (𝑞 <Q 𝑦 → ∃𝑟Q (𝑞 <Q 𝑟𝑟 <Q 𝑦))
5 simpll 527 . . . . . . . 8 (((𝑞 <Q 𝑟𝑟 <Q 𝑦) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑞 <Q 𝑟)
6 19.8a 1590 . . . . . . . . . 10 ((𝑟 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
71recexprlemell 7618 . . . . . . . . . 10 (𝑟 ∈ (1st𝐵) ↔ ∃𝑦(𝑟 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
86, 7sylibr 134 . . . . . . . . 9 ((𝑟 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑟 ∈ (1st𝐵))
98adantll 476 . . . . . . . 8 (((𝑞 <Q 𝑟𝑟 <Q 𝑦) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑟 ∈ (1st𝐵))
105, 9jca 306 . . . . . . 7 (((𝑞 <Q 𝑟𝑟 <Q 𝑦) ∧ (*Q𝑦) ∈ (2nd𝐴)) → (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))
1110expcom 116 . . . . . 6 ((*Q𝑦) ∈ (2nd𝐴) → ((𝑞 <Q 𝑟𝑟 <Q 𝑦) → (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵))))
1211reximdv 2578 . . . . 5 ((*Q𝑦) ∈ (2nd𝐴) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 <Q 𝑦) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵))))
134, 12mpan9 281 . . . 4 ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))
1413exlimiv 1598 . . 3 (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))
152, 14sylbi 121 . 2 (𝑞 ∈ (1st𝐵) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))
16153ad2ant3 1020 1 ((𝐴P𝑞Q𝑞 ∈ (1st𝐵)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wrex 2456  cop 3595   class class class wbr 4002  cfv 5215  1st c1st 6136  2nd c2nd 6137  Qcnq 7276  *Qcrq 7280   <Q cltq 7281  Pcnp 7287
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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
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 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-eprel 4288  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-1o 6414  df-oadd 6418  df-omul 6419  df-er 6532  df-ec 6534  df-qs 6538  df-ni 7300  df-pli 7301  df-mi 7302  df-lti 7303  df-plpq 7340  df-mpq 7341  df-enq 7343  df-nqqs 7344  df-plqqs 7345  df-mqqs 7346  df-1nqqs 7347  df-rq 7348  df-ltnqqs 7349
This theorem is referenced by:  recexprlemrnd  7625
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