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Mirrors > Home > MPE Home > Th. List > hashprlei | Structured version Visualization version GIF version |
Description: An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
hashprlei | ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4325 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | hashsnlei 13419 | . 2 ⊢ ({𝐴} ∈ Fin ∧ (♯‘{𝐴}) ≤ 1) | |
3 | hashsnlei 13419 | . 2 ⊢ ({𝐵} ∈ Fin ∧ (♯‘{𝐵}) ≤ 1) | |
4 | 1nn0 11521 | . 2 ⊢ 1 ∈ ℕ0 | |
5 | 1p1e2 11347 | . 2 ⊢ (1 + 1) = 2 | |
6 | 1, 2, 3, 4, 4, 5 | hashunlei 13425 | 1 ⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 2140 {csn 4322 {cpr 4324 class class class wbr 4805 ‘cfv 6050 Fincfn 8124 1c1 10150 ≤ cle 10288 2c2 11283 ♯chash 13332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-card 8976 df-cda 9203 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-nn 11234 df-2 11292 df-n0 11506 df-xnn0 11577 df-z 11591 df-uz 11901 df-fz 12541 df-hash 13333 |
This theorem is referenced by: hashle2pr 13472 hashtplei 13479 hashtpg 13480 upgrbi 26209 upgr1elem 26228 kur14lem8 31524 |
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