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Mirrors > Home > MPE Home > Th. List > ensymb | Structured version Visualization version GIF version |
Description: Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ensymb | ⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ener 8002 | . . . 4 ⊢ ≈ Er V | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ≈ Er V) |
3 | 2 | ersymb 7756 | . 2 ⊢ (⊤ → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) |
4 | 3 | trud 1493 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ⊤wtru 1484 Vcvv 3200 class class class wbr 4653 Er wer 7739 ≈ 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-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-er 7742 df-en 7956 |
This theorem is referenced by: ensym 8005 0sdomg 8089 snnen2o 8149 cantnfp1lem2 8576 cantnflem1 8586 iscard2 8802 dffin1-5 9210 pmtrsn 17939 volmeas 30294 isnumbasgrplem1 37671 rp-isfinite6 37864 |
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