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Mirrors > Home > ILE Home > Th. List > regexmidlemm | GIF version |
Description: Lemma for regexmid 4313. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
Ref | Expression |
---|---|
regexmidlemm.a | ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} |
Ref | Expression |
---|---|
regexmidlemm | ⊢ ∃𝑦 𝑦 ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p0ex 3986 | . . . 4 ⊢ {∅} ∈ V | |
2 | 1 | prid2 3523 | . . 3 ⊢ {∅} ∈ {∅, {∅}} |
3 | eqid 2083 | . . . 4 ⊢ {∅} = {∅} | |
4 | 3 | orci 683 | . . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)) |
5 | eqeq1 2089 | . . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅})) | |
6 | eqeq1 2089 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
7 | 6 | anbi1d 453 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑))) |
8 | 5, 7 | orbi12d 740 | . . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
9 | regexmidlemm.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} | |
10 | 8, 9 | elrab2 2762 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
11 | 2, 4, 10 | mpbir2an 884 | . 2 ⊢ {∅} ∈ 𝐴 |
12 | elex2 2626 | . 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 7 | 1 ⊢ ∃𝑦 𝑦 ∈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ∨ wo 662 = wceq 1285 ∃wex 1422 ∈ wcel 1434 {crab 2357 ∅c0 3269 {csn 3422 {cpr 3423 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rab 2362 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 |
This theorem is referenced by: regexmid 4313 reg2exmid 4314 reg3exmid 4357 nnregexmid 4396 |
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