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Mirrors > Home > MPE Home > Th. List > ecexr | Structured version Visualization version GIF version |
Description: A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecexr | ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 3879 | . . 3 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ¬ (𝑅 “ {𝐵}) = ∅) | |
2 | snprc 4197 | . . . . 5 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅) | |
3 | imaeq2 5381 | . . . . 5 ⊢ ({𝐵} = ∅ → (𝑅 “ {𝐵}) = (𝑅 “ ∅)) | |
4 | 2, 3 | sylbi 206 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = (𝑅 “ ∅)) |
5 | ima0 5400 | . . . 4 ⊢ (𝑅 “ ∅) = ∅ | |
6 | 4, 5 | syl6eq 2660 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝑅 “ {𝐵}) = ∅) |
7 | 1, 6 | nsyl2 141 | . 2 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → 𝐵 ∈ V) |
8 | df-ec 7631 | . 2 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
9 | 7, 8 | eleq2s 2706 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 “ cima 5041 [cec 7627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 |
This theorem is referenced by: relelec 7674 ecdmn0 7676 erdisj 7681 |
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