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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 4668 | . . 3 ⊢ 𝑥 ∈ {𝑥} | |
2 | vsnex 5440 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq2 2828 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥})) | |
4 | eleq2 2828 | . . . . 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 4647 | . . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
9 | equcomi 2014 | . . 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 1535 = wceq 1537 ∈ wcel 2106 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 |
This theorem is referenced by: (None) |
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