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Mirrors > Home > ILE Home > Th. List > acexmidlemb | GIF version |
Description: Lemma for acexmid 5867. (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 |
---|---|
acexmidlemb | ⊢ (∅ ∈ 𝐵 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acexmidlem.b | . . . 4 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} | |
2 | 1 | eleq2i 2244 | . . 3 ⊢ (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}) |
3 | 0ex 4127 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | prid1 3697 | . . . 4 ⊢ ∅ ∈ {∅, {∅}} |
5 | eqeq1 2184 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅})) | |
6 | 5 | orbi1d 791 | . . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑))) |
7 | 6 | elrab3 2894 | . . . 4 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)) |
9 | 2, 8 | bitri 184 | . 2 ⊢ (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑)) |
10 | noel 3426 | . . . 4 ⊢ ¬ ∅ ∈ ∅ | |
11 | 3 | snid 3622 | . . . . 5 ⊢ ∅ ∈ {∅} |
12 | eleq2 2241 | . . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅})) | |
13 | 11, 12 | mpbiri 168 | . . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅) |
14 | 10, 13 | mto 662 | . . 3 ⊢ ¬ ∅ = {∅} |
15 | orel1 725 | . . 3 ⊢ (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑)) | |
16 | 14, 15 | ax-mp 5 | . 2 ⊢ ((∅ = {∅} ∨ 𝜑) → 𝜑) |
17 | 9, 16 | sylbi 121 | 1 ⊢ (∅ ∈ 𝐵 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 {crab 2459 ∅c0 3422 {csn 3591 {cpr 3592 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-nul 4126 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-nul 3423 df-sn 3597 df-pr 3598 |
This theorem is referenced by: acexmidlem1 5864 |
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