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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 8535 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | sdomnen 8537 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
3 | 2 | con2i 141 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐴 ≺ 𝐵) |
4 | 1, 3 | jca 514 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
5 | brdom2 8538 | . . . 4 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
6 | 5 | biimpi 218 | . . 3 ⊢ (𝐴 ≼ 𝐵 → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) |
7 | 6 | orcanai 999 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵) → 𝐴 ≈ 𝐵) |
8 | 4, 7 | impbii 211 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 class class class wbr 5065 ≈ cen 8505 ≼ cdom 8506 ≺ csdm 8507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-f1o 6361 df-en 8509 df-dom 8510 df-sdom 8511 |
This theorem is referenced by: marypha1lem 8896 tskwe 9378 infxpenlem 9438 cdainflem 9612 axcclem 9878 alephsuc3 10001 gchen1 10046 gchen2 10047 inatsk 10199 ufilen 22537 dirith2 26103 f1ocnt 30524 lindsenlbs 34886 mblfinlem1 34928 axccdom 41485 axccd2 41494 |
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