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Mirrors > Home > ILE Home > Th. List > enpr1g | 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 3541 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 6691 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
3 | 1, 2 | eqbrtrrid 3964 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 {csn 3527 {cpr 3528 class class class wbr 3929 1oc1o 6306 ≈ cen 6632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-1o 6313 df-en 6635 |
This theorem is referenced by: pr2ne 7048 |
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