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Mirrors > Home > MPE Home > Th. List > entr4i | Structured version Visualization version GIF version |
Description: A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
Ref | Expression |
---|---|
entr4i.1 | ⊢ 𝐴 ≈ 𝐵 |
entr4i.2 | ⊢ 𝐶 ≈ 𝐵 |
Ref | Expression |
---|---|
entr4i | ⊢ 𝐴 ≈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | entr4i.1 | . 2 ⊢ 𝐴 ≈ 𝐵 | |
2 | entr4i.2 | . . 3 ⊢ 𝐶 ≈ 𝐵 | |
3 | 2 | ensymi 8985 | . 2 ⊢ 𝐵 ≈ 𝐶 |
4 | 1, 3 | entri 8989 | 1 ⊢ 𝐴 ≈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5142 ≈ cen 8921 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-er 8688 df-en 8925 |
This theorem is referenced by: fodomfi 9310 xpnnen 16138 rpnnen 16154 rexpen 16155 cnso 16174 |
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