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Mirrors > Home > ILE Home > Th. List > suplocexprlemml | GIF version |
Description: Lemma for suplocexpr 7712. 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 7702 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ P) |
5 | 4 | sselda 3155 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ P) |
6 | prop 7462 | . . . . 5 ⊢ (𝑥 ∈ P → 〈(1st ‘𝑥), (2nd ‘𝑥)〉 ∈ P) | |
7 | prml 7464 | . . . . 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 2550 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝑥)) |
10 | r19.2m 3509 | . . 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 7700 | . . . 4 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥)) | |
13 | 12 | rexbii 2484 | . . 3 ⊢ (∃𝑠 ∈ Q 𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ Q ∃𝑥 ∈ 𝐴 𝑠 ∈ (1st ‘𝑥)) |
14 | rexcom 2641 | . . 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)) |
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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