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Mirrors > Home > MPE Home > Th. List > Mathboxes > fisshasheq | Structured version Visualization version GIF version |
Description: A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.) |
Ref | Expression |
---|---|
fisshasheq | ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfi 8735 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ Fin) | |
2 | 1 | 3adant3 1127 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 ∈ Fin) |
3 | hashen 13705 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) | |
4 | 3 | biimp3a 1464 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 ≈ 𝐵) |
5 | pm3.2 472 | . . . . . . . . 9 ⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → (𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵))) | |
6 | 5 | 3ad2ant2 1129 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵)) → (𝐴 ⊆ 𝐵 → (𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵))) |
7 | fisseneq 8726 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵) | |
8 | 7 | 3expa 1113 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵) ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵) |
9 | 8 | expcom 416 | . . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵) → 𝐴 = 𝐵)) |
10 | 4, 6, 9 | sylsyld 61 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵)) → (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) |
11 | 10 | 3expb 1115 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵))) → (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵)) |
12 | 11 | expcom 416 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵)) → (𝐴 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵))) |
13 | 12 | com23 86 | . . . 4 ⊢ ((𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵)) → (𝐴 ⊆ 𝐵 → (𝐴 ∈ Fin → 𝐴 = 𝐵))) |
14 | 13 | 3impia 1112 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ (♯‘𝐴) = (♯‘𝐵) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∈ Fin → 𝐴 = 𝐵)) |
15 | 14 | 3com23 1121 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → (𝐴 ∈ Fin → 𝐴 = 𝐵)) |
16 | 2, 15 | mpd 15 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ⊆ wss 3933 class class class wbr 5063 ‘cfv 6352 ≈ cen 8503 Fincfn 8506 ♯chash 13688 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-n0 11896 df-z 11980 df-uz 12242 df-hash 13689 |
This theorem is referenced by: cusgredgex 32392 |
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