Theorem suplocexprlemml 7776

Description: Lemma for suplocexpr 7785. 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 7775	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ P)
5	4	sselda 3179	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ P)
6		prop 7535	. . . . 5 ⊢ (𝑥 ∈ P → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ P)
7		prml 7537	. . . . 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 2567	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
10		r19.2m 3533	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥)) → ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
11	1, 9, 10	syl2anc 411	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
12		suplocexprlemell 7773	. . . 4 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥))
13	12	rexbii 2501	. . 3 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥))
14		rexcom 2658	. . 3 ⊢ (∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥) ↔ ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
15	13, 14	bitri 184	. 2 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥))
16	11, 15	sylibr 134	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ⟨cop 3621  ∪ cuni 3835   class class class wbr 4029   “ cima 4662  ‘cfv 5254  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340  Pcnp 7351  <_P cltp 7355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-qs 6593  df-ni 7364  df-nqqs 7408  df-inp 7526  df-iltp 7530

This theorem is referenced by:  suplocexprlemex  7782