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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 2539 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
2 | abss 3960 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 ∈ {𝑦})) | |
3 | velsn 4543 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
4 | 3 | imbi2i 339 | . . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ {𝑦}) ↔ (𝜑 → 𝑥 = 𝑦)) |
5 | 4 | albii 1827 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
6 | 2, 5 | bitri 278 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
7 | snex 5309 | . . . . 5 ⊢ {𝑦} ∈ V | |
8 | 7 | ssex 5199 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
9 | 6, 8 | sylbir 238 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V) |
10 | 9 | exlimiv 1938 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V) |
11 | 1, 10 | sylbi 220 | 1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 ∈ wcel 2112 ∃*wmo 2537 {cab 2714 Vcvv 3398 ⊆ wss 3853 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 df-pr 4530 |
This theorem is referenced by: rmorabex 5329 euabex 5330 satfv0 32987 |
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