![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4633 | . 2 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 9015 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
3 | 1, 2 | eqbrtrrid 5174 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 {csn 4620 {cpr 4622 class class class wbr 5138 1oc1o 8454 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-suc 6360 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-1o 8461 df-en 8936 |
This theorem is referenced by: pr2neOLD 9996 |
Copyright terms: Public domain | W3C validator |