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 36017 | . . 3 ⊢ (Prt 𝐴 → ∼ Er ∪ 𝐴) |
3 | erexb 8316 | . . 3 ⊢ ( ∼ Er ∪ 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ ∪ 𝐴 ∈ V)) |
5 | uniexb 7488 | . 2 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
6 | 4, 5 | syl6bbr 291 | 1 ⊢ (Prt 𝐴 → ( ∼ ∈ V ↔ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ∪ cuni 4840 {copab 5130 Er wer 8288 Prt wprt 36009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-er 8291 df-prt 36010 |
This theorem is referenced by: (None) |
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