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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 9045 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ 𝐴 ≈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5148 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-er 8744 df-en 8985 |
This theorem is referenced by: entr2i 9048 entr3i 9049 entr4i 9050 infxpenc2 10060 cfpwsdom 10622 hashxplem 14469 xpnnen 16244 qnnen 16246 rpnnen 16260 rexpen 16261 odhash 19607 cygctb 19925 met2ndci 24551 re2ndc 24837 iscmet3 25341 dyadmbl 25649 opnmblALT 25652 mbfimaopnlem 25704 aannenlem3 26387 mblfinlem1 37644 heiborlem3 37800 heibor 37808 irrapx1 42816 zenom 44992 qenom 45311 |
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