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Mirrors > Home > ILE Home > Th. List > elrabi | GIF version |
Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
Ref | Expression |
---|---|
elrabi | ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clelab 2296 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑))) | |
2 | eleq1 2233 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉)) | |
3 | 2 | anbi1d 462 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ (𝐴 ∈ 𝑉 ∧ 𝜑))) |
4 | 3 | simprbda 381 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉) |
5 | 4 | exlimiv 1591 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ (𝑥 ∈ 𝑉 ∧ 𝜑)) → 𝐴 ∈ 𝑉) |
6 | 1, 5 | sylbi 120 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} → 𝐴 ∈ 𝑉) |
7 | df-rab 2457 | . 2 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} | |
8 | 6, 7 | eleq2s 2265 | 1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 {crab 2452 |
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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-rab 2457 |
This theorem is referenced by: ordtriexmidlem 4503 ordtri2or2exmidlem 4510 onsucelsucexmidlem 4513 ordsoexmid 4546 reg3exmidlemwe 4563 elfvmptrab1 5590 acexmidlemcase 5848 ssfirab 6911 exmidonfinlem 7170 cc4f 7231 genpelvl 7474 genpelvu 7475 suplocsrlempr 7769 nnindnn 7855 sup3exmid 8873 nnind 8894 supinfneg 9554 infsupneg 9555 supminfex 9556 ublbneg 9572 hashinfuni 10711 zsupcllemstep 11900 infssuzex 11904 infssuzledc 11905 bezoutlemsup 11964 uzwodc 11992 lcmgcdlem 12031 phisum 12194 oddennn 12347 evenennn 12348 znnen 12353 ennnfonelemg 12358 txdis1cn 13072 reopnap 13332 divcnap 13349 limccl 13422 dvlemap 13443 dvaddxxbr 13459 dvmulxxbr 13460 dvcoapbr 13465 dvcjbr 13466 dvrecap 13471 dveflem 13481 |
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