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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 8563 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ 𝐴 ≈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5068 ≈ cen 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-er 8291 df-en 8512 |
This theorem is referenced by: entr2i 8566 entr3i 8567 entr4i 8568 infxpenc2 9450 cfpwsdom 10008 hashxplem 13797 xpnnen 15566 qnnen 15568 rpnnen 15582 rexpen 15583 odhash 18701 cygctb 19014 met2ndci 23134 re2ndc 23411 iscmet3 23898 dyadmbl 24203 opnmblALT 24206 mbfimaopnlem 24258 aannenlem3 24921 mblfinlem1 34931 heiborlem3 35093 heibor 35101 irrapx1 39432 zenom 41321 qenom 41636 |
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