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Mirrors > Home > ILE Home > Th. List > onsucelsucexmidlem1 | GIF version |
Description: Lemma for onsucelsucexmid 4215. (Contributed by Jim Kingdon, 2-Aug-2019.) |
Ref | Expression |
---|---|
onsucelsucexmidlem1 | ⊢ ∅ ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3875 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | prid1 3467 | . 2 ⊢ ∅ ∈ {∅, {∅}} |
3 | eqid 2037 | . . 3 ⊢ ∅ = ∅ | |
4 | 3 | orci 649 | . 2 ⊢ (∅ = ∅ ∨ φ) |
5 | eqeq1 2043 | . . . 4 ⊢ (x = ∅ → (x = ∅ ↔ ∅ = ∅)) | |
6 | 5 | orbi1d 704 | . . 3 ⊢ (x = ∅ → ((x = ∅ ∨ φ) ↔ (∅ = ∅ ∨ φ))) |
7 | 6 | elrab 2692 | . 2 ⊢ (∅ ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} ↔ (∅ ∈ {∅, {∅}} ∧ (∅ = ∅ ∨ φ))) |
8 | 2, 4, 7 | mpbir2an 848 | 1 ⊢ ∅ ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 628 = wceq 1242 ∈ wcel 1390 {crab 2304 ∅c0 3218 {csn 3367 {cpr 3368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-nul 3219 df-sn 3373 df-pr 3374 |
This theorem is referenced by: onsucelsucexmidlem 4214 onsucelsucexmid 4215 acexmidlem2 5452 |
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