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Mirrors > Home > MPE Home > Th. List > en1uniel | Structured version Visualization version GIF version |
Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
en1uniel | ⊢ (𝑆 ≈ 1𝑜 → ∪ 𝑆 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 7846 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelexi 5082 | . . 3 ⊢ (𝑆 ≈ 1𝑜 → 𝑆 ∈ V) |
3 | uniexg 6853 | . . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V) | |
4 | snidg 4153 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝑆 ≈ 1𝑜 → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | en1b 7910 | . . 3 ⊢ (𝑆 ≈ 1𝑜 ↔ 𝑆 = {∪ 𝑆}) | |
7 | 6 | biimpi 205 | . 2 ⊢ (𝑆 ≈ 1𝑜 → 𝑆 = {∪ 𝑆}) |
8 | 5, 7 | eleqtrrd 2691 | 1 ⊢ (𝑆 ≈ 1𝑜 → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ∪ cuni 4372 class class class wbr 4583 1𝑜c1o 7440 ≈ cen 7838 |
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-8 1979 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 ax-un 6847 |
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-reu 2903 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-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-1o 7447 df-en 7842 |
This theorem is referenced by: en2eleq 8714 en2other2 8715 pmtrf 17698 pmtrmvd 17699 pmtrfinv 17704 frgpcyg 19741 |
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