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| Mirrors > Home > MPE Home > Th. List > enfii | Structured version Visualization version GIF version | ||
| Description: A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| enfii | ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isfi 9016 | . . . . . 6 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) | |
| 2 | df-rex 3071 | . . . . . 6 ⊢ (∃𝑥 ∈ ω 𝐵 ≈ 𝑥 ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥)) | |
| 3 | 1, 2 | sylbb 219 | . . . . 5 ⊢ (𝐵 ∈ Fin → ∃𝑥(𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥)) |
| 4 | ensymfib 9224 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (𝐵 ≈ 𝐴 ↔ 𝐴 ≈ 𝐵)) | |
| 5 | 4 | biimparc 479 | . . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐵 ≈ 𝐴) |
| 6 | 19.41v 1949 | . . . . . 6 ⊢ (∃𝑥((𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥) ∧ 𝐵 ≈ 𝐴) ↔ (∃𝑥(𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥) ∧ 𝐵 ≈ 𝐴)) | |
| 7 | simp1 1137 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → 𝑥 ∈ ω) | |
| 8 | nnfi 9207 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
| 9 | ensymfib 9224 | . . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ 𝐵 ↔ 𝐵 ≈ 𝑥)) | |
| 10 | 9 | biimpar 477 | . . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ 𝑥) → 𝑥 ≈ 𝐵) |
| 11 | 10 | 3adant3 1133 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → 𝑥 ≈ 𝐵) |
| 12 | entrfil 9225 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Fin ∧ 𝑥 ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → 𝑥 ≈ 𝐴) | |
| 13 | 11, 12 | syld3an2 1413 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → 𝑥 ≈ 𝐴) |
| 14 | ensymfib 9224 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ Fin → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)) | |
| 15 | 14 | 3ad2ant1 1134 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → (𝑥 ≈ 𝐴 ↔ 𝐴 ≈ 𝑥)) |
| 16 | 13, 15 | mpbid 232 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → 𝐴 ≈ 𝑥) |
| 17 | 8, 16 | syl3an1 1164 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → 𝐴 ≈ 𝑥) |
| 18 | 7, 17 | jca 511 | . . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥 ∧ 𝐵 ≈ 𝐴) → (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) |
| 19 | 18 | 3expa 1119 | . . . . . . 7 ⊢ (((𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥) ∧ 𝐵 ≈ 𝐴) → (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) |
| 20 | 19 | eximi 1835 | . . . . . 6 ⊢ (∃𝑥((𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥) ∧ 𝐵 ≈ 𝐴) → ∃𝑥(𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) |
| 21 | 6, 20 | sylbir 235 | . . . . 5 ⊢ ((∃𝑥(𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥) ∧ 𝐵 ≈ 𝐴) → ∃𝑥(𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) |
| 22 | 3, 5, 21 | syl2an2 686 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∃𝑥(𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) |
| 23 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃𝑥(𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) | |
| 24 | 22, 23 | sylibr 234 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
| 25 | isfi 9016 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
| 26 | 24, 25 | sylibr 234 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
| 27 | 26 | ancoms 458 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ωcom 7887 ≈ cen 8982 Fincfn 8985 |
| 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-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-fin 8989 |
| This theorem is referenced by: enfi 9227 domfi 9229 entrfi 9230 entrfir 9231 domsdomtrfi 9242 f1finf1o 9305 en1eqsnOLD 9309 isfinite2 9334 xpfiOLD 9359 fofinf1o 9372 cnvfiALT 9379 f1dmvrnfibi 9381 cantnfcl 9707 en2eqpr 10047 fzfi 14013 hasheni 14387 fz1isolem 14500 isercolllem2 15702 isercoll 15704 summolem2 15752 zsum 15754 prodmolem2 15971 zprod 15973 bitsf1 16483 simpgnsgd 20120 ovoliunlem1 25537 wlksnfi 29927 eupthfi 30224 eulerpartlemgs2 34382 derangenlem 35176 erdsze2lem2 35209 heicant 37662 sticksstones18 42165 sticksstones19 42166 |
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