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Mirrors > Home > ILE Home > Th. List > rext | GIF version |
Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.) |
Ref | Expression |
---|---|
rext | ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 3402 | . . 3 ⊢ 𝑥 ∈ {𝑥} |
3 | snexgOLD 3935 | . . . . 5 ⊢ (𝑥 ∈ V → {𝑥} ∈ V) | |
4 | 1, 3 | ax-mp 7 | . . . 4 ⊢ {𝑥} ∈ V |
5 | eleq2 2101 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥})) | |
6 | eleq2 2101 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥})) | |
7 | 5, 6 | imbi12d 223 | . . . 4 ⊢ (𝑧 = {𝑥} → ((𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))) |
8 | 4, 7 | spcv 2646 | . . 3 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})) |
9 | 2, 8 | mpi 15 | . 2 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑦 ∈ {𝑥}) |
10 | velsn 3392 | . . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
11 | equcomi 1592 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
12 | 10, 11 | sylbi 114 | . 2 ⊢ (𝑦 ∈ {𝑥} → 𝑥 = 𝑦) |
13 | 9, 12 | syl 14 | 1 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 |
This theorem is referenced by: (None) |
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