Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ensymb | 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 6757 | . . . 4 ⊢ ≈ Er V | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → ≈ Er V) |
3 | 2 | ersymb 6527 | . 2 ⊢ (⊤ → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) |
4 | 3 | mptru 1357 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ⊤wtru 1349 Vcvv 2730 class class class wbr 3989 Er wer 6510 ≈ cen 6716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-er 6513 df-en 6719 |
This theorem is referenced by: ensym 6759 php5 6836 snnen2og 6837 snnen2oprc 6838 phpeqd 6910 |
Copyright terms: Public domain | W3C validator |