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| Description: The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| prtlem18.1 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | 
| Ref | Expression | 
|---|---|
| prtex | ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | prtlem18.1 | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | |
| 2 | 1 | prter1 38880 | . . 3 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴) | 
| 3 | erexb 8770 | . . 3 ⊢ ( ∼ Er ∪ 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) | 
| 5 | uniexb 7784 | . 2 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
| 6 | 4, 5 | bitr4di 289 | 1 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∪ cuni 4907 {copab 5205 Er wer 8742 Prt wprt 38872 | 
| 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-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-er 8745 df-prt 38873 | 
| This theorem is referenced by: (None) | 
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