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Mirrors > Home > ILE Home > Th. List > acexmidlemph | 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 |
---|---|
acexmidlemph | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 664 | . . . 4 ⊢ (𝜑 → (𝑥 = ∅ ∨ 𝜑)) | |
2 | 1 | ralrimivw 2435 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑)) |
3 | acexmidlem.a | . . . . 5 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} | |
4 | 3 | eqeq2i 2091 | . . . 4 ⊢ ({∅, {∅}} = 𝐴 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) |
5 | rabid2 2530 | . . . 4 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑)) | |
6 | 4, 5 | bitri 182 | . . 3 ⊢ ({∅, {∅}} = 𝐴 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑)) |
7 | 2, 6 | sylibr 132 | . 2 ⊢ (𝜑 → {∅, {∅}} = 𝐴) |
8 | olc 664 | . . . 4 ⊢ (𝜑 → (𝑥 = {∅} ∨ 𝜑)) | |
9 | 8 | ralrimivw 2435 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑)) |
10 | acexmidlem.b | . . . . 5 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} | |
11 | 10 | eqeq2i 2091 | . . . 4 ⊢ ({∅, {∅}} = 𝐵 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}) |
12 | rabid2 2530 | . . . 4 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑)) | |
13 | 11, 12 | bitri 182 | . . 3 ⊢ ({∅, {∅}} = 𝐵 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑)) |
14 | 9, 13 | sylibr 132 | . 2 ⊢ (𝜑 → {∅, {∅}} = 𝐵) |
15 | 7, 14 | eqtr3d 2115 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 661 = wceq 1284 ∀wral 2348 {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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-ral 2353 df-rab 2357 |
This theorem is referenced by: acexmidlemab 5526 |
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