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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 8161 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | |
4 | 1, 2, 3 | mp2an 672 | 1 ⊢ 𝐴 ≈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4786 ≈ cen 8106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-er 7896 df-en 8110 |
This theorem is referenced by: entr2i 8164 entr3i 8165 entr4i 8166 infxpenc2 9045 cfpwsdom 9608 hashxplem 13422 xpnnen 15145 qnnen 15148 rpnnen 15162 rexpen 15163 odhash 18196 cygctb 18500 met2ndci 22547 re2ndc 22824 iscmet3 23310 dyadmbl 23588 opnmblALT 23591 mbfimaopnlem 23642 aannenlem3 24305 mblfinlem1 33779 heiborlem3 33944 heibor 33952 irrapx1 37918 zenom 39740 qenom 40093 |
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