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Mirrors > Home > MPE Home > Th. List > fisseneq | Structured version Visualization version GIF version |
Description: A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.) |
Ref | Expression |
---|---|
fisseneq | ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pss 3885 | . . . . . 6 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | pssinf 8888 | . . . . . . 7 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ∈ Fin) | |
3 | 2 | expcom 417 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ⊊ 𝐵 → ¬ 𝐵 ∈ Fin)) |
4 | 1, 3 | syl5bir 246 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → ¬ 𝐵 ∈ Fin)) |
5 | 4 | expdimp 456 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ Fin)) |
6 | 5 | necon4ad 2959 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∈ Fin → 𝐴 = 𝐵)) |
7 | 6 | 3impia 1119 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 = 𝐵) |
8 | 7 | 3com13 1126 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ⊆ wss 3866 ⊊ wpss 3867 class class class wbr 5053 ≈ cen 8623 Fincfn 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-om 7645 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 |
This theorem is referenced by: en1eqsn 8904 en2eqpr 9621 en2eleq 9622 psgnunilem1 18885 sylow2blem1 19009 fislw 19014 sylow2 19015 cyggenod 19268 ablfac1c 19458 ablfac1eu 19460 fta1blem 25066 vieta1 25205 upgrex 27183 fisshasheq 32786 poimirlem26 35540 fiuneneq 40725 |
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