![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > regexmidlemm | GIF version |
Description: Lemma for regexmid 4552. 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 4206 | . . . 4 ⊢ {∅} ∈ V | |
2 | 1 | prid2 3714 | . . 3 ⊢ {∅} ∈ {∅, {∅}} |
3 | eqid 2189 | . . . 4 ⊢ {∅} = {∅} | |
4 | 3 | orci 732 | . . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)) |
5 | eqeq1 2196 | . . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅})) | |
6 | eqeq1 2196 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
7 | 6 | anbi1d 465 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑))) |
8 | 5, 7 | orbi12d 794 | . . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
9 | regexmidlemm.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} | |
10 | 8, 9 | elrab2 2911 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
11 | 2, 4, 10 | mpbir2an 944 | . 2 ⊢ {∅} ∈ 𝐴 |
12 | elex2 2768 | . 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 5 | 1 ⊢ ∃𝑦 𝑦 ∈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∨ wo 709 = wceq 1364 ∃wex 1503 ∈ wcel 2160 {crab 2472 ∅c0 3437 {csn 3607 {cpr 3608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 |
This theorem is referenced by: regexmid 4552 reg2exmid 4553 reg3exmid 4597 nnregexmid 4638 |
Copyright terms: Public domain | W3C validator |