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Mirrors > Home > ILE Home > Th. List > entr | GIF version |
Description: Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.) |
Ref | Expression |
---|---|
entr | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ener 6673 | . . . 4 ⊢ ≈ Er V | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → ≈ Er V) |
3 | 2 | ertr 6444 | . 2 ⊢ (⊤ → ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)) |
4 | 3 | mptru 1340 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊤wtru 1332 Vcvv 2686 class class class wbr 3929 Er wer 6426 ≈ cen 6632 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-er 6429 df-en 6635 |
This theorem is referenced by: entri 6680 en2sn 6707 xpsnen2g 6723 enen1 6734 enen2 6735 ssenen 6745 phplem4 6749 snnen2og 6753 php5dom 6757 phplem4on 6761 dif1en 6773 dif1enen 6774 fisbth 6777 diffisn 6787 unsnfidcex 6808 unsnfidcel 6809 f1finf1o 6835 en1eqsn 6836 endjusym 6981 carden2bex 7045 pm54.43 7046 pr2ne 7048 djuen 7067 djuenun 7068 djuassen 7073 frecfzen2 10200 uzennn 10209 hashunlem 10550 hashxp 10572 1nprm 11795 hashdvds 11897 unennn 11910 ennnfonelemen 11934 ennnfonelemim 11937 exmidunben 11939 ctinfom 11941 ctinf 11943 pwf1oexmid 13194 |
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