![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > moabex | Structured version Visualization version GIF version |
Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.) |
Ref | Expression |
---|---|
moabex | ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2526 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
2 | abss 4050 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 ∈ {𝑦})) | |
3 | velsn 4637 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
4 | 3 | imbi2i 336 | . . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ {𝑦}) ↔ (𝜑 → 𝑥 = 𝑦)) |
5 | 4 | albii 1813 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
6 | 2, 5 | bitri 275 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
7 | vsnex 5420 | . . . . 5 ⊢ {𝑦} ∈ V | |
8 | 7 | ssex 5312 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
9 | 6, 8 | sylbir 234 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V) |
10 | 9 | exlimiv 1925 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V) |
11 | 1, 10 | sylbi 216 | 1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ∃*wmo 2524 {cab 2701 Vcvv 3466 ⊆ wss 3941 {csn 4621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-rab 3425 df-v 3468 df-un 3946 df-in 3948 df-ss 3958 df-sn 4622 df-pr 4624 |
This theorem is referenced by: rmorabex 5451 euabex 5452 satfv0 34840 |
Copyright terms: Public domain | W3C validator |