Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ensn1g | Structured version Visualization version 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 4187 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 4663 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜)) |
3 | vex 3203 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 8020 | . 2 ⊢ {𝑥} ≈ 1𝑜 |
5 | 2, 4 | vtoclg 3266 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {csn 4177 class class class wbr 4653 1𝑜c1o 7553 ≈ cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-suc 5729 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-1o 7560 df-en 7956 |
This theorem is referenced by: enpr1g 8022 en1b 8024 en2sn 8037 snfi 8038 snnen2o 8149 sucxpdom 8169 en1eqsn 8190 en1eqsnbi 8191 pr2nelem 8827 prdom2 8829 cda1en 8997 snct 29491 rngoueqz 33739 |
Copyright terms: Public domain | W3C validator |