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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 9000 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | sdomnen 9002 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
3 | 2 | con2i 139 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐴 ≺ 𝐵) |
4 | 1, 3 | jca 510 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
5 | brdom2 9003 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
6 | 5 | biimpi 215 | . . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
7 | 6 | orcanai 1000 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵) → 𝐴 ≈ 𝐵) |
8 | 4, 7 | impbii 208 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∨ wo 845 class class class wbr 5149 ≈ cen 8961 ≼ cdom 8962 ≺ csdm 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-f1o 6556 df-en 8965 df-dom 8966 df-sdom 8967 |
This theorem is referenced by: marypha1lem 9458 tskwe 9975 infxpenlem 10038 cdainflem 10212 axcclem 10482 alephsuc3 10605 gchen1 10650 gchen2 10651 inatsk 10803 ufilen 23878 dirith2 27506 f1ocnt 32652 lindsenlbs 37219 mblfinlem1 37261 axccdom 44734 axccd2 44742 |
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