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| Mirrors > Home > MPE Home > Th. List > enpr1g | Structured version Visualization version GIF version | ||
| Description: {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.) |
| Ref | Expression |
|---|---|
| enpr1g | ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfsn2 4604 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
| 2 | ensn1g 8995 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
| 3 | 1, 2 | eqbrtrrid 5145 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2109 {csn 4591 {cpr 4593 class class class wbr 5109 1oc1o 8429 ≈ cen 8917 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-mo 2534 df-clab 2709 df-cleq 2722 df-clel 2804 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-br 5110 df-opab 5172 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-suc 6340 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-1o 8436 df-en 8921 |
| This theorem is referenced by: pr2neOLD 9964 |
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