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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 8637 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵) | |
2 | endom 8536 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | ensym 8558 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | endom 8536 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
6 | 2, 5 | jca 514 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) |
7 | 1, 6 | impbii 211 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 class class class wbr 5066 ≈ cen 8506 ≼ cdom 8507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-er 8289 df-en 8510 df-dom 8511 |
This theorem is referenced by: sbthcl 8639 dom0 8645 carden2 9416 axgroth2 10247 |
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