![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > bren2 | Structured version Visualization version GIF version |
Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
bren2 | ⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 8519 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | sdomnen 8521 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
3 | 2 | con2i 141 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐴 ≺ 𝐵) |
4 | 1, 3 | jca 515 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
5 | brdom2 8522 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
6 | 5 | biimpi 219 | . . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
7 | 6 | orcanai 1000 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵) → 𝐴 ≈ 𝐵) |
8 | 4, 7 | impbii 212 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 844 class class class wbr 5030 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 |
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-pr 5295 |
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-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-f1o 6331 df-en 8493 df-dom 8494 df-sdom 8495 |
This theorem is referenced by: marypha1lem 8881 tskwe 9363 infxpenlem 9424 cdainflem 9598 axcclem 9868 alephsuc3 9991 gchen1 10036 gchen2 10037 inatsk 10189 ufilen 22535 dirith2 26112 f1ocnt 30551 lindsenlbs 35052 mblfinlem1 35094 axccdom 41853 axccd2 41862 |
Copyright terms: Public domain | W3C validator |