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Mirrors > Home > ILE Home > Th. List > domentr | GIF version |
Description: Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
Ref | Expression |
---|---|
domentr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 6625 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | domtr 6647 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan2 284 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 class class class wbr 3899 ≈ cen 6600 ≼ cdom 6601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-f1o 5100 df-en 6603 df-dom 6604 |
This theorem is referenced by: xpdom1g 6695 domen2 6705 phplem4dom 6724 phpm 6727 fisbth 6745 infnfi 6757 fientri3 6771 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 hashennnuni 10493 xpct 11836 pwf1oexmid 13121 sbthom 13148 |
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