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Mathbox for Rodolfo Medina |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtex | Structured version Visualization version GIF version |
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 38243 | . . 3 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴) |
3 | erexb 8725 | . . 3 ⊢ ( ∼ Er ∪ 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) |
5 | uniexb 7745 | . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 Vcvv 3466 ∪ cuni 4900 {copab 5201 Er wer 8697 Prt wprt 38235 |
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-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-er 8700 df-prt 38236 |
This theorem is referenced by: (None) |
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