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Mirrors > Home > ILE Home > Th. List > domen1 | GIF version |
Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Ref | Expression |
---|---|
domen1 | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐶 ↔ 𝐵 ≼ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensym 6780 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | endomtr 6789 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≼ 𝐶) → 𝐵 ≼ 𝐶) | |
3 | 1, 2 | sylan 283 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ≼ 𝐶) → 𝐵 ≼ 𝐶) |
4 | endomtr 6789 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
5 | 3, 4 | impbida 596 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐶 ↔ 𝐵 ≼ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 class class class wbr 4003 ≈ cen 6737 ≼ cdom 6738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-er 6534 df-en 6740 df-dom 6741 |
This theorem is referenced by: fihashdom 10778 |
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