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Mirrors > Home > ILE Home > Th. List > suplocexprlemml | GIF version |
Description: Lemma for suplocexpr 7533. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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))) |
Ref | Expression |
---|---|
suplocexprlemml | ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴)) |
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)) |
Colors of variables: wff set class |
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 1st c1st 6036 2nd 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 |
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