Theorem suplocexprlemml 7524

Description: Lemma for suplocexpr 7533. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))

Assertion

Ref	Expression
suplocexprlemml	⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))

Distinct variable groups:   𝐴,𝑠,𝑥,𝑦   𝜑,𝑠,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)

Proof of Theorem suplocexprlemml

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
2		suplocexpr.ub	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
3		suplocexpr.loc	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
4	1, 2, 3	suplocexprlemss 7523	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ P)
5	4	sselda 3097	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ P)
6		prop 7283	. . . . 5 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
7		prml 7285	. . . . 5 ⊢ (⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
8	5, 6, 7	3syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
9	8	ralrimiva 2505	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
10		r19.2m 3449	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥)) → ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
11	1, 9, 10	syl2anc 408	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
12		suplocexprlemell 7521	. . . 4 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥))
13	12	rexbii 2442	. . 3 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥))
14		rexcom 2595	. . 3 ⊢ (∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
15	13, 14	bitri 183	. 2 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
16	11, 15	sylibr 133	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  ⟨cop 3530  ∪ cuni 3736   class class class wbr 3929   “ cima 4542  ‘cfv 5123  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088  Pcnp 7099  <_P cltp 7103

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-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 7112  df-nqqs 7156  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocexprlemex  7530