Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > en2sn | Structured version Visualization version GIF version |
Description: Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) |
Ref | Expression |
---|---|
en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensn1g 8021 | . 2 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1𝑜) | |
2 | ensn1g 8021 | . . 3 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1𝑜) | |
3 | 2 | ensymd 8007 | . 2 ⊢ (𝐵 ∈ 𝐷 → 1𝑜 ≈ {𝐵}) |
4 | entr 8008 | . 2 ⊢ (({𝐴} ≈ 1𝑜 ∧ 1𝑜 ≈ {𝐵}) → {𝐴} ≈ {𝐵}) | |
5 | 1, 3, 4 | syl2an 494 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ 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-pow 4843 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-pw 4160 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-res 5126 df-ima 5127 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-er 7742 df-en 7956 |
This theorem is referenced by: difsnen 8042 domunsncan 8060 domunsn 8110 limensuci 8136 infensuc 8138 sucdom2 8156 dif1en 8193 dif1card 8833 fin23lem26 9147 unsnen 9375 canthp1lem1 9474 fzennn 12767 hashsng 13159 mreexexlem4d 16307 |
Copyright terms: Public domain | W3C validator |