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Mirrors > Home > MPE Home > Th. List > hashprb | Structured version Visualization version GIF version |
Description: The size of an unordered pair is 2 if and only if its elements are different sets. (Contributed by Alexander van der Vekens, 17-Jan-2018.) |
Ref | Expression |
---|---|
hashprb | ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁) ↔ (#‘{𝑀, 𝑁}) = 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashprg 13182 | . . 3 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 ≠ 𝑁 ↔ (#‘{𝑀, 𝑁}) = 2)) | |
2 | 1 | biimp3a 1432 | . 2 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁) → (#‘{𝑀, 𝑁}) = 2) |
3 | elprchashprn2 13184 | . . . 4 ⊢ (¬ 𝑀 ∈ V → ¬ (#‘{𝑀, 𝑁}) = 2) | |
4 | pm2.21 120 | . . . 4 ⊢ (¬ (#‘{𝑀, 𝑁}) = 2 → ((#‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (¬ 𝑀 ∈ V → ((#‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁))) |
6 | elprchashprn2 13184 | . . . 4 ⊢ (¬ 𝑁 ∈ V → ¬ (#‘{𝑁, 𝑀}) = 2) | |
7 | prcom 4267 | . . . . . . 7 ⊢ {𝑁, 𝑀} = {𝑀, 𝑁} | |
8 | 7 | fveq2i 6194 | . . . . . 6 ⊢ (#‘{𝑁, 𝑀}) = (#‘{𝑀, 𝑁}) |
9 | 8 | eqeq1i 2627 | . . . . 5 ⊢ ((#‘{𝑁, 𝑀}) = 2 ↔ (#‘{𝑀, 𝑁}) = 2) |
10 | 9, 4 | sylnbi 320 | . . . 4 ⊢ (¬ (#‘{𝑁, 𝑀}) = 2 → ((#‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁))) |
11 | 6, 10 | syl 17 | . . 3 ⊢ (¬ 𝑁 ∈ V → ((#‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁))) |
12 | simpll 790 | . . . . 5 ⊢ (((𝑀 ∈ V ∧ 𝑁 ∈ V) ∧ (#‘{𝑀, 𝑁}) = 2) → 𝑀 ∈ V) | |
13 | simplr 792 | . . . . 5 ⊢ (((𝑀 ∈ V ∧ 𝑁 ∈ V) ∧ (#‘{𝑀, 𝑁}) = 2) → 𝑁 ∈ V) | |
14 | 1 | biimpar 502 | . . . . 5 ⊢ (((𝑀 ∈ V ∧ 𝑁 ∈ V) ∧ (#‘{𝑀, 𝑁}) = 2) → 𝑀 ≠ 𝑁) |
15 | 12, 13, 14 | 3jca 1242 | . . . 4 ⊢ (((𝑀 ∈ V ∧ 𝑁 ∈ V) ∧ (#‘{𝑀, 𝑁}) = 2) → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁)) |
16 | 15 | ex 450 | . . 3 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ((#‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁))) |
17 | 5, 11, 16 | ecase 983 | . 2 ⊢ ((#‘{𝑀, 𝑁}) = 2 → (𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁)) |
18 | 2, 17 | impbii 199 | 1 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V ∧ 𝑀 ≠ 𝑁) ↔ (#‘{𝑀, 𝑁}) = 2) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 {cpr 4179 ‘cfv 5888 2c2 11070 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: hashprdifel 13186 prsshashgt1 13198 symg2hash 17817 cplgr2vpr 26329 |
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