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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 8975 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | sdomnen 8977 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
3 | 2 | con2i 139 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐴 ≺ 𝐵) |
4 | 1, 3 | jca 513 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
5 | brdom2 8978 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
6 | 5 | biimpi 215 | . . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
7 | 6 | orcanai 1002 | . 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 397 ∨ wo 846 class class class wbr 5149 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-f1o 6551 df-en 8940 df-dom 8941 df-sdom 8942 |
This theorem is referenced by: marypha1lem 9428 tskwe 9945 infxpenlem 10008 cdainflem 10182 axcclem 10452 alephsuc3 10575 gchen1 10620 gchen2 10621 inatsk 10773 ufilen 23434 dirith2 27031 f1ocnt 32013 lindsenlbs 36483 mblfinlem1 36525 axccdom 43921 axccd2 43929 |
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