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Mirrors > Home > ILE Home > Th. List > acexmidlema | GIF version |
Description: Lemma for acexmid 5766. (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 2204 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) |
3 | p0ex 4107 | . . . . 5 ⊢ {∅} ∈ V | |
4 | 3 | prid2 3625 | . . . 4 ⊢ {∅} ∈ {∅, {∅}} |
5 | eqeq1 2144 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
6 | 5 | orbi1d 780 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑))) |
7 | 6 | elrab3 2836 | . . . 4 ⊢ ({∅} ∈ {∅, {∅}} → ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑))) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ ({∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ({∅} = ∅ ∨ 𝜑)) |
9 | 2, 8 | bitri 183 | . 2 ⊢ ({∅} ∈ 𝐴 ↔ ({∅} = ∅ ∨ 𝜑)) |
10 | noel 3362 | . . . 4 ⊢ ¬ ∅ ∈ ∅ | |
11 | 0ex 4050 | . . . . . 6 ⊢ ∅ ∈ V | |
12 | 11 | snid 3551 | . . . . 5 ⊢ ∅ ∈ {∅} |
13 | eleq2 2201 | . . . . 5 ⊢ ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅)) | |
14 | 12, 13 | mpbii 147 | . . . 4 ⊢ ({∅} = ∅ → ∅ ∈ ∅) |
15 | 10, 14 | mto 651 | . . 3 ⊢ ¬ {∅} = ∅ |
16 | orel1 714 | . . 3 ⊢ (¬ {∅} = ∅ → (({∅} = ∅ ∨ 𝜑) → 𝜑)) | |
17 | 15, 16 | ax-mp 5 | . 2 ⊢ (({∅} = ∅ ∨ 𝜑) → 𝜑) |
18 | 9, 17 | sylbi 120 | 1 ⊢ ({∅} ∈ 𝐴 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 {crab 2418 ∅c0 3358 {csn 3522 {cpr 3523 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 |
This theorem is referenced by: acexmidlem1 5763 |
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