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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 4666 | . . 3 ⊢ 𝑥 ∈ {𝑥} | |
2 | vsnex 5430 | . . . 4 ⊢ {𝑥} ∈ V | |
3 | eleq2 2823 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥})) | |
4 | eleq2 2823 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥})) | |
5 | 3, 4 | imbi12d 345 | . . . 4 ⊢ (𝑧 = {𝑥} → ((𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))) |
6 | 2, 5 | spcv 3596 | . . 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 4645 | . . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
9 | equcomi 2021 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
10 | 8, 9 | sylbi 216 | . 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 1540 = wceq 1542 ∈ wcel 2107 {csn 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3954 df-sn 4630 df-pr 4632 |
This theorem is referenced by: (None) |
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