![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sdomen1 | Structured version Visualization version GIF version |
Description: Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.) |
Ref | Expression |
---|---|
sdomen1 | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≺ 𝐶 ↔ 𝐵 ≺ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensym 8357 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | ensdomtr 8451 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≺ 𝐶) → 𝐵 ≺ 𝐶) | |
3 | 1, 2 | sylan 572 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ≺ 𝐶) → 𝐵 ≺ 𝐶) |
4 | ensdomtr 8451 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | |
5 | 3, 4 | impbida 788 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≺ 𝐶 ↔ 𝐵 ≺ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 class class class wbr 4930 ≈ cen 8305 ≺ csdm 8307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 |
This theorem is referenced by: isfiniteg 8575 djufi 9412 alephval2 9794 engch 9850 |
Copyright terms: Public domain | W3C validator |