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Theorem suplocexprlemml 7703
Description: Lemma for suplocexpr 7712. The lower cut of the putative supremum is inhabited. (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
suplocexprlemml (𝜑 → ∃𝑠Q 𝑠 (1st𝐴))
Distinct variable groups:   𝐴,𝑠,𝑥,𝑦   𝜑,𝑠,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)

Proof of Theorem suplocexprlemml
StepHypRef Expression
1 suplocexpr.m . . 3 (𝜑 → ∃𝑥 𝑥𝐴)
2 suplocexpr.ub . . . . . . 7 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
3 suplocexpr.loc . . . . . . 7 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
41, 2, 3suplocexprlemss 7702 . . . . . 6 (𝜑𝐴P)
54sselda 3155 . . . . 5 ((𝜑𝑥𝐴) → 𝑥P)
6 prop 7462 . . . . 5 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
7 prml 7464 . . . . 5 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ P → ∃𝑠Q 𝑠 ∈ (1st𝑥))
85, 6, 73syl 17 . . . 4 ((𝜑𝑥𝐴) → ∃𝑠Q 𝑠 ∈ (1st𝑥))
98ralrimiva 2550 . . 3 (𝜑 → ∀𝑥𝐴𝑠Q 𝑠 ∈ (1st𝑥))
10 r19.2m 3509 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴𝑠Q 𝑠 ∈ (1st𝑥)) → ∃𝑥𝐴𝑠Q 𝑠 ∈ (1st𝑥))
111, 9, 10syl2anc 411 . 2 (𝜑 → ∃𝑥𝐴𝑠Q 𝑠 ∈ (1st𝑥))
12 suplocexprlemell 7700 . . . 4 (𝑠 (1st𝐴) ↔ ∃𝑥𝐴 𝑠 ∈ (1st𝑥))
1312rexbii 2484 . . 3 (∃𝑠Q 𝑠 (1st𝐴) ↔ ∃𝑠Q𝑥𝐴 𝑠 ∈ (1st𝑥))
14 rexcom 2641 . . 3 (∃𝑠Q𝑥𝐴 𝑠 ∈ (1st𝑥) ↔ ∃𝑥𝐴𝑠Q 𝑠 ∈ (1st𝑥))
1513, 14bitri 184 . 2 (∃𝑠Q 𝑠 (1st𝐴) ↔ ∃𝑥𝐴𝑠Q 𝑠 ∈ (1st𝑥))
1611, 15sylibr 134 1 (𝜑 → ∃𝑠Q 𝑠 (1st𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  wex 1492  wcel 2148  wral 2455  wrex 2456  cop 3594   cuni 3807   class class class wbr 4000  cima 4626  cfv 5212  1st c1st 6133  2nd c2nd 6134  Qcnq 7267  Pcnp 7278  <P cltp 7282
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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-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-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-qs 6535  df-ni 7291  df-nqqs 7335  df-inp 7453  df-iltp 7457
This theorem is referenced by:  suplocexprlemex  7709
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