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| Mirrors > Home > MPE Home > Th. List > sbthb | Structured version Visualization version GIF version | ||
| Description: Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.) |
| Ref | Expression |
|---|---|
| sbthb | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbth 9133 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | |
| 2 | endom 9019 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
| 3 | ensym 9043 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
| 4 | endom 9019 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
| 6 | 2, 5 | jca 511 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) |
| 7 | 1, 6 | impbii 209 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-er 8745 df-en 8986 df-dom 8987 |
| This theorem is referenced by: sbthcl 9135 dom0OLD 9143 carden2 10027 axgroth2 10865 |
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