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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 4538 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 8557 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
3 | 1, 2 | eqbrtrrid 5066 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 {csn 4525 {cpr 4527 class class class wbr 5030 1oc1o 8078 ≈ cen 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-1o 8085 df-en 8493 |
This theorem is referenced by: pr2ne 9416 |
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