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Mirrors > Home > ILE Home > Th. List > acexmidlema | GIF version |
Description: Lemma for acexmid 5836. (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 2231 | . . 3 ⊢ ({∅} ∈ 𝐴 ↔ {∅} ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) |
3 | p0ex 4162 | . . . . 5 ⊢ {∅} ∈ V | |
4 | 3 | prid2 3678 | . . . 4 ⊢ {∅} ∈ {∅, {∅}} |
5 | eqeq1 2171 | . . . . . 6 ⊢ (𝑥 = {∅} → (𝑥 = ∅ ↔ {∅} = ∅)) | |
6 | 5 | orbi1d 781 | . . . . 5 ⊢ (𝑥 = {∅} → ((𝑥 = ∅ ∨ 𝜑) ↔ ({∅} = ∅ ∨ 𝜑))) |
7 | 6 | elrab3 2879 | . . . 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 3409 | . . . 4 ⊢ ¬ ∅ ∈ ∅ | |
11 | 0ex 4104 | . . . . . 6 ⊢ ∅ ∈ V | |
12 | 11 | snid 3602 | . . . . 5 ⊢ ∅ ∈ {∅} |
13 | eleq2 2228 | . . . . 5 ⊢ ({∅} = ∅ → (∅ ∈ {∅} ↔ ∅ ∈ ∅)) | |
14 | 12, 13 | mpbii 147 | . . . 4 ⊢ ({∅} = ∅ → ∅ ∈ ∅) |
15 | 10, 14 | mto 652 | . . 3 ⊢ ¬ {∅} = ∅ |
16 | orel1 715 | . . 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 698 = wceq 1342 ∈ wcel 2135 {crab 2446 ∅c0 3405 {csn 3571 {cpr 3572 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rab 2451 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 |
This theorem is referenced by: acexmidlem1 5833 |
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