![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ensn1g | GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.) |
Ref | Expression |
---|---|
ensn1g | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3411 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 3797 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜)) |
3 | vex 2605 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 6335 | . 2 ⊢ {𝑥} ≈ 1𝑜 |
5 | 2, 4 | vtoclg 2659 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 {csn 3400 class class class wbr 3787 1𝑜c1o 6052 ≈ cen 6278 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3898 ax-nul 3906 ax-pow 3950 ax-pr 3966 ax-un 4190 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3253 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-br 3788 df-opab 3842 df-id 4050 df-suc 4128 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-rn 4376 df-fun 4928 df-fn 4929 df-f 4930 df-f1 4931 df-fo 4932 df-f1o 4933 df-1o 6059 df-en 6281 |
This theorem is referenced by: enpr1g 6337 en1bg 6339 en2sn 6349 snfig 6350 snnen2og 6384 pr2nelem 6509 |
Copyright terms: Public domain | W3C validator |