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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 6772 | . . . 4 ⊢ ≈ Er V | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → ≈ Er V) |
3 | 2 | ertr 6543 | . 2 ⊢ (⊤ → ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)) |
4 | 3 | mptru 1362 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ⊤wtru 1354 Vcvv 2737 class class class wbr 4000 Er wer 6525 ≈ cen 6731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-er 6528 df-en 6734 |
This theorem is referenced by: entri 6779 en2sn 6806 xpsnen2g 6822 enen1 6833 enen2 6834 ssenen 6844 phplem4 6848 snnen2og 6852 php5dom 6856 phplem4on 6860 dif1en 6872 dif1enen 6873 fisbth 6876 diffisn 6886 unsnfidcex 6912 unsnfidcel 6913 f1finf1o 6939 en1eqsn 6940 endjusym 7088 carden2bex 7181 pm54.43 7182 pr2ne 7184 djuen 7203 djuenun 7204 djuassen 7209 frecfzen2 10400 uzennn 10409 hashunlem 10755 hashxp 10777 1nprm 12084 hashdvds 12191 unennn 12368 ennnfonelemen 12392 ennnfonelemim 12395 exmidunben 12397 ctinfom 12399 ctinf 12401 pwf1oexmid 14371 |
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