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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 8347 | . . . 4 ⊢ ≈ Er V | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ≈ Er V) |
3 | 2 | ersymb 8097 | . 2 ⊢ (⊤ → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) |
4 | 3 | mptru 1514 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ⊤wtru 1508 Vcvv 3409 class class class wbr 4923 Er wer 8080 ≈ cen 8297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-er 8083 df-en 8301 |
This theorem is referenced by: ensym 8349 0sdomg 8436 snnen2o 8496 cantnfp1lem2 8930 cantnflem1 8940 iscard2 9193 dffin1-5 9602 pmtrsn 18403 volmeas 31135 isnumbasgrplem1 39097 rp-isfinite6 39280 prproropen 43038 |
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