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Mirrors > Home > ILE Home > Th. List > acexmidlema | GIF version |
Description: Lemma for acexmid 5454. (Contributed by Jim Kingdon, 6-Aug-2019.) |
Ref | Expression |
---|---|
acexmidlem.a | ⊢ A = {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} |
acexmidlem.b | ⊢ B = {x ∈ {∅, {∅}} ∣ (x = {∅} ∨ φ)} |
acexmidlem.c | ⊢ 𝐶 = {A, B} |
Ref | Expression |
---|---|
acexmidlema | ⊢ ({∅} ∈ A → φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acexmidlem.a | . . . 4 ⊢ A = {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} | |
2 | 1 | eleq2i 2101 | . . 3 ⊢ ({∅} ∈ A ↔ {∅} ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)}) |
3 | p0ex 3930 | . . . . 5 ⊢ {∅} ∈ V | |
4 | 3 | prid2 3468 | . . . 4 ⊢ {∅} ∈ {∅, {∅}} |
5 | eqeq1 2043 | . . . . . 6 ⊢ (x = {∅} → (x = ∅ ↔ {∅} = ∅)) | |
6 | 5 | orbi1d 704 | . . . . 5 ⊢ (x = {∅} → ((x = ∅ ∨ φ) ↔ ({∅} = ∅ ∨ φ))) |
7 | 6 | elrab3 2693 | . . . 4 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} ↔ ({∅} = ∅ ∨ φ))) |
8 | 4, 7 | ax-mp 7 | . . 3 ⊢ ({∅} ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} ↔ ({∅} = ∅ ∨ φ)) |
9 | 2, 8 | bitri 173 | . 2 ⊢ ({∅} ∈ A ↔ ({∅} = ∅ ∨ φ)) |
10 | noel 3222 | . . . 4 ⊢ ¬ ∅ ∈ ∅ | |
11 | 0ex 3875 | . . . . . 6 ⊢ ∅ ∈ V | |
12 | 11 | snid 3394 | . . . . 5 ⊢ ∅ ∈ {∅} |
13 | eleq2 2098 | . . . . 5 ⊢ ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅)) | |
14 | 12, 13 | mpbii 136 | . . . 4 ⊢ ({∅} = ∅ → ∅ ∈ ∅) |
15 | 10, 14 | mto 587 | . . 3 ⊢ ¬ {∅} = ∅ |
16 | orel1 643 | . . 3 ⊢ (¬ {∅} = ∅ → (({∅} = ∅ ∨ φ) → φ)) | |
17 | 15, 16 | ax-mp 7 | . 2 ⊢ (({∅} = ∅ ∨ φ) → φ) |
18 | 9, 17 | sylbi 114 | 1 ⊢ ({∅} ∈ A → φ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ wo 628 = wceq 1242 ∈ wcel 1390 {crab 2304 ∅c0 3218 {csn 3367 {cpr 3368 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 |
This theorem is referenced by: acexmidlem1 5451 |
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