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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 | vsnid 3624 | . . 3 ⊢ 𝑥 ∈ {𝑥} | |
2 | vex 2740 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | snex 4184 | . . . 4 ⊢ {𝑥} ∈ V |
4 | eleq2 2241 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥})) | |
5 | eleq2 2241 | . . . . 5 ⊢ (𝑧 = {𝑥} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥})) | |
6 | 4, 5 | imbi12d 234 | . . . 4 ⊢ (𝑧 = {𝑥} → ((𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))) |
7 | 3, 6 | spcv 2831 | . . 3 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})) |
8 | 1, 7 | mpi 15 | . 2 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑦 ∈ {𝑥}) |
9 | velsn 3609 | . . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥) | |
10 | equcomi 1704 | . . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
11 | 9, 10 | sylbi 121 | . 2 ⊢ (𝑦 ∈ {𝑥} → 𝑥 = 𝑦) |
12 | 8, 11 | syl 14 | 1 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 = wceq 1353 ∈ wcel 2148 {csn 3592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 |
This theorem is referenced by: (None) |
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