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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem400 | Structured version Visualization version GIF version |
Description: Lemma for prter2 33184 and also a property of partitions . (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
prtlem13.1 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} |
Ref | Expression |
---|---|
prtlem400 | ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neirr 2791 | . 2 ⊢ ¬ ∅ ≠ ∅ | |
2 | prtlem13.1 | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | |
3 | 2 | prtlem16 33172 | . . 3 ⊢ dom ∼ = ∪ 𝐴 |
4 | elqsn0 7703 | . . 3 ⊢ ((dom ∼ = ∪ 𝐴 ∧ ∅ ∈ (∪ 𝐴 / ∼ )) → ∅ ≠ ∅) | |
5 | 3, 4 | mpan 702 | . 2 ⊢ (∅ ∈ (∪ 𝐴 / ∼ ) → ∅ ≠ ∅) |
6 | 1, 5 | mto 187 | 1 ⊢ ¬ ∅ ∈ (∪ 𝐴 / ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∅c0 3874 ∪ cuni 4372 {copab 4642 dom cdm 5038 / cqs 7628 |
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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-uni 4373 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 df-qs 7635 |
This theorem is referenced by: prter2 33184 |
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