![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > domen2 | Structured version Visualization version GIF version |
Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Ref | Expression |
---|---|
domen2 | ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domentr 8551 | . . 3 ⊢ ((𝐶 ≼ 𝐴 ∧ 𝐴 ≈ 𝐵) → 𝐶 ≼ 𝐵) | |
2 | 1 | ancoms 462 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼ 𝐴) → 𝐶 ≼ 𝐵) |
3 | ensym 8541 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | domentr 8551 | . . . 4 ⊢ ((𝐶 ≼ 𝐵 ∧ 𝐵 ≈ 𝐴) → 𝐶 ≼ 𝐴) | |
5 | 4 | ancoms 462 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐶 ≼ 𝐵) → 𝐶 ≼ 𝐴) |
6 | 3, 5 | sylan 583 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼ 𝐵) → 𝐶 ≼ 𝐴) |
7 | 2, 6 | impbida 800 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 class class class wbr 5030 ≈ cen 8489 ≼ cdom 8490 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-er 8272 df-en 8493 df-dom 8494 |
This theorem is referenced by: infdiffi 9105 carddomi2 9383 numdom 9449 djudom2 9594 infdif 9620 fin45 9803 fin67 9806 aleph1 9982 gchdomtri 10040 gchpwdom 10081 gchhar 10090 ctbnfien 39759 |
Copyright terms: Public domain | W3C validator |