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Mirrors > Home > MPE Home > Th. List > entri | Structured version Visualization version GIF version |
Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
Ref | Expression |
---|---|
entri.1 | ⊢ 𝐴 ≈ 𝐵 |
entri.2 | ⊢ 𝐵 ≈ 𝐶 |
Ref | Expression |
---|---|
entri | ⊢ 𝐴 ≈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | entri.1 | . 2 ⊢ 𝐴 ≈ 𝐵 | |
2 | entri.2 | . 2 ⊢ 𝐵 ≈ 𝐶 | |
3 | entr 8544 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ 𝐴 ≈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5030 ≈ cen 8489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-er 8272 df-en 8493 |
This theorem is referenced by: entr2i 8547 entr3i 8548 entr4i 8549 infxpenc2 9433 cfpwsdom 9995 hashxplem 13790 xpnnen 15556 qnnen 15558 rpnnen 15572 rexpen 15573 odhash 18691 cygctb 19005 met2ndci 23129 re2ndc 23406 iscmet3 23897 dyadmbl 24204 opnmblALT 24207 mbfimaopnlem 24259 aannenlem3 24926 mblfinlem1 35094 heiborlem3 35251 heibor 35259 irrapx1 39769 zenom 41686 qenom 41993 |
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