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Mirrors > Home > ILE Home > Th. List > acexmidlemb | GIF version |
Description: Lemma for acexmid 5531. (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 2145 | . . 3 ⊢ (∅ ∈ 𝐵 ↔ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}) |
3 | 0ex 3905 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 3 | prid1 3498 | . . . 4 ⊢ ∅ ∈ {∅, {∅}} |
5 | eqeq1 2087 | . . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = {∅} ↔ ∅ = {∅})) | |
6 | 5 | orbi1d 737 | . . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 = {∅} ∨ 𝜑) ↔ (∅ = {∅} ∨ 𝜑))) |
7 | 6 | elrab3 2750 | . . . 4 ⊢ (∅ ∈ {∅, {∅}} → (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑))) |
8 | 4, 7 | ax-mp 7 | . . 3 ⊢ (∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ (∅ = {∅} ∨ 𝜑)) |
9 | 2, 8 | bitri 182 | . 2 ⊢ (∅ ∈ 𝐵 ↔ (∅ = {∅} ∨ 𝜑)) |
10 | noel 3255 | . . . 4 ⊢ ¬ ∅ ∈ ∅ | |
11 | 3 | snid 3425 | . . . . 5 ⊢ ∅ ∈ {∅} |
12 | eleq2 2142 | . . . . 5 ⊢ (∅ = {∅} → (∅ ∈ ∅ ↔ ∅ ∈ {∅})) | |
13 | 11, 12 | mpbiri 166 | . . . 4 ⊢ (∅ = {∅} → ∅ ∈ ∅) |
14 | 10, 13 | mto 620 | . . 3 ⊢ ¬ ∅ = {∅} |
15 | orel1 676 | . . 3 ⊢ (¬ ∅ = {∅} → ((∅ = {∅} ∨ 𝜑) → 𝜑)) | |
16 | 14, 15 | ax-mp 7 | . 2 ⊢ ((∅ = {∅} ∨ 𝜑) → 𝜑) |
17 | 9, 16 | sylbi 119 | 1 ⊢ (∅ ∈ 𝐵 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 {crab 2352 ∅c0 3251 {csn 3398 {cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-nul 3252 df-sn 3404 df-pr 3405 |
This theorem is referenced by: acexmidlem1 5528 |
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