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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 2540 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
| 2 | abss 4063 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 ∈ {𝑦})) | |
| 3 | velsn 4642 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
| 4 | 3 | imbi2i 336 | . . . . . 6 ⊢ ((𝜑 → 𝑥 ∈ {𝑦}) ↔ (𝜑 → 𝑥 = 𝑦)) |
| 5 | 4 | albii 1819 | . . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
| 6 | 2, 5 | bitri 275 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
| 7 | vsnex 5434 | . . . . 5 ⊢ {𝑦} ∈ V | |
| 8 | 7 | ssex 5321 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑦} → {𝑥 ∣ 𝜑} ∈ V) |
| 9 | 6, 8 | sylbir 235 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V) |
| 10 | 9 | exlimiv 1930 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V) |
| 11 | 1, 10 | sylbi 217 | 1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 {cab 2714 Vcvv 3480 ⊆ wss 3951 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: rmorabex 5465 euabex 5466 satfv0 35363 |
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