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Mirrors > Home > MPE Home > Th. List > hashen1 | Structured version Visualization version GIF version |
Description: A set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) |
Ref | Expression |
---|---|
hashen1 | ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5213 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | hashsng 13733 | . . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (♯‘{∅}) = 1 |
4 | 3 | eqcomi 2832 | . . . 4 ⊢ 1 = (♯‘{∅}) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 1 = (♯‘{∅})) |
6 | 5 | eqeq2d 2834 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = (♯‘{∅}))) |
7 | simpr 487 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → (♯‘𝐴) = (♯‘{∅})) | |
8 | 1nn0 11916 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
9 | 3, 8 | eqeltri 2911 | . . . . . . . 8 ⊢ (♯‘{∅}) ∈ ℕ0 |
10 | hashvnfin 13724 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘{∅}) ∈ ℕ0) → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) | |
11 | 9, 10 | mpan2 689 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) |
12 | 11 | imp 409 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ∈ Fin) |
13 | snfi 8596 | . . . . . 6 ⊢ {∅} ∈ Fin | |
14 | hashen 13710 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {∅} ∈ Fin) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) | |
15 | 12, 13, 14 | sylancl 588 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
16 | 7, 15 | mpbid 234 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ≈ {∅}) |
17 | 16 | ex 415 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ≈ {∅})) |
18 | hasheni 13711 | . . 3 ⊢ (𝐴 ≈ {∅} → (♯‘𝐴) = (♯‘{∅})) | |
19 | 17, 18 | impbid1 227 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
20 | df1o2 8118 | . . . . 5 ⊢ 1o = {∅} | |
21 | 20 | eqcomi 2832 | . . . 4 ⊢ {∅} = 1o |
22 | 21 | breq2i 5076 | . . 3 ⊢ (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o) |
23 | 22 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o)) |
24 | 6, 19, 23 | 3bitrd 307 | 1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {csn 4569 class class class wbr 5068 ‘cfv 6357 1oc1o 8097 ≈ cen 8508 Fincfn 8511 1c1 10540 ℕ0cn0 11900 ♯chash 13693 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: hash1elsn 13735 euhash1 13784 0ring 20045 0ring01eqbi 20048 lfuhgr3 32368 spthcycl 32378 |
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