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Mirrors > Home > ILE Home > Th. List > regexmidlemm | GIF version |
Description: Lemma for regexmid 4278. 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 3959 | . . . 4 ⊢ {∅} ∈ V | |
2 | 1 | prid2 3499 | . . 3 ⊢ {∅} ∈ {∅, {∅}} |
3 | eqid 2081 | . . . 4 ⊢ {∅} = {∅} | |
4 | 3 | orci 682 | . . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)) |
5 | eqeq1 2087 | . . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅})) | |
6 | eqeq1 2087 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
7 | 6 | anbi1d 452 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑))) |
8 | 5, 7 | orbi12d 739 | . . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
9 | regexmidlemm.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} | |
10 | 8, 9 | elrab2 2751 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
11 | 2, 4, 10 | mpbir2an 883 | . 2 ⊢ {∅} ∈ 𝐴 |
12 | elex2 2615 | . 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 661 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {crab 2352 ∅c0 3251 {csn 3398 {cpr 3399 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 |
This theorem is referenced by: regexmid 4278 reg2exmid 4279 reg3exmid 4322 nnregexmid 4360 |
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