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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 38345 | . . 3 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴) |
3 | erexb 8743 | . . 3 ⊢ ( ∼ Er ∪ 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) |
5 | uniexb 7760 | . 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 1534 ∈ wcel 2099 ∃wrex 3066 Vcvv 3470 ∪ cuni 4903 {copab 5204 Er wer 8715 Prt wprt 38337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-er 8718 df-prt 38338 |
This theorem is referenced by: (None) |
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