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Mirrors > Home > ILE Home > Th. List > regexmidlemm | GIF version |
Description: Lemma for regexmid 4260. 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 3939 | . . . 4 ⊢ {∅} ∈ V | |
2 | 1 | prid2 3477 | . . 3 ⊢ {∅} ∈ {∅, {∅}} |
3 | eqid 2040 | . . . 4 ⊢ {∅} = {∅} | |
4 | 3 | orci 650 | . . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)) |
5 | eqeq1 2046 | . . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅})) | |
6 | eqeq1 2046 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
7 | 6 | anbi1d 438 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑))) |
8 | 5, 7 | orbi12d 707 | . . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
9 | regexmidlemm.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} | |
10 | 8, 9 | elrab2 2700 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
11 | 2, 4, 10 | mpbir2an 849 | . 2 ⊢ {∅} ∈ 𝐴 |
12 | elex2 2570 | . 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 7 | 1 ⊢ ∃𝑦 𝑦 ∈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∨ wo 629 = wceq 1243 ∃wex 1381 ∈ wcel 1393 {crab 2310 ∅c0 3224 {csn 3375 {cpr 3376 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 |
This theorem is referenced by: regexmid 4260 reg2exmid 4261 reg3exmid 4304 nnregexmid 4342 |
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