Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > regexmidlemm | GIF version |
Description: Lemma for regexmid 4450. 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 4112 | . . . 4 ⊢ {∅} ∈ V | |
2 | 1 | prid2 3630 | . . 3 ⊢ {∅} ∈ {∅, {∅}} |
3 | eqid 2139 | . . . 4 ⊢ {∅} = {∅} | |
4 | 3 | orci 720 | . . 3 ⊢ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)) |
5 | eqeq1 2146 | . . . . 5 ⊢ (𝑥 = {∅} → (𝑥 = {∅} ↔ {∅} = {∅})) | |
6 | eqeq1 2146 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
7 | 6 | anbi1d 460 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∧ 𝜑) ↔ ({∅} = ∅ ∧ 𝜑))) |
8 | 5, 7 | orbi12d 782 | . . . 4 ⊢ (𝑥 = {∅} → ((𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑)) ↔ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
9 | regexmidlemm.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} | |
10 | 8, 9 | elrab2 2843 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} ∈ {∅, {∅}} ∧ ({∅} = {∅} ∨ ({∅} = ∅ ∧ 𝜑)))) |
11 | 2, 4, 10 | mpbir2an 926 | . 2 ⊢ {∅} ∈ 𝐴 |
12 | elex2 2702 | . 2 ⊢ ({∅} ∈ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) | |
13 | 11, 12 | ax-mp 5 | 1 ⊢ ∃𝑦 𝑦 ∈ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∨ wo 697 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {crab 2420 ∅c0 3363 {csn 3527 {cpr 3528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 |
This theorem is referenced by: regexmid 4450 reg2exmid 4451 reg3exmid 4494 nnregexmid 4534 |
Copyright terms: Public domain | W3C validator |