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| Mirrors > Home > MPE Home > Th. List > rext | Structured version Visualization version 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 | vsnid 4663 | . . 3 ⊢ 𝑥 ∈ {𝑥} | |
| 2 | vsnex 5434 | . . . 4 ⊢ {𝑥} ∈ V | |
| 3 | eleq2 2830 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥})) | |
| 4 | eleq2 2830 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥})) | |
| 5 | 3, 4 | imbi12d 344 | . . . 4 ⊢ (𝑧 = {𝑥} → ((𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))) |
| 6 | 2, 5 | spcv 3605 | . . 3 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})) |
| 7 | 1, 6 | mpi 20 | . 2 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑦 ∈ {𝑥}) |
| 8 | velsn 4642 | . . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
| 9 | equcomi 2016 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
| 10 | 8, 9 | sylbi 217 | . 2 ⊢ (𝑦 ∈ {𝑥} → 𝑥 = 𝑦) |
| 11 | 7, 10 | syl 17 | 1 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: (None) |
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