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Mirrors > Home > MPE Home > Th. List > elprchashprn2 | Structured version Visualization version GIF version |
Description: If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.) |
Ref | Expression |
---|---|
elprchashprn2 | ⊢ (¬ 𝑀 ∈ V → ¬ (♯‘{𝑀, 𝑁}) = 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prprc1 4703 | . 2 ⊢ (¬ 𝑀 ∈ V → {𝑀, 𝑁} = {𝑁}) | |
2 | hashsng 13733 | . . . 4 ⊢ (𝑁 ∈ V → (♯‘{𝑁}) = 1) | |
3 | fveq2 6672 | . . . . . . . . 9 ⊢ ({𝑀, 𝑁} = {𝑁} → (♯‘{𝑀, 𝑁}) = (♯‘{𝑁})) | |
4 | 3 | eqcomd 2829 | . . . . . . . 8 ⊢ ({𝑀, 𝑁} = {𝑁} → (♯‘{𝑁}) = (♯‘{𝑀, 𝑁})) |
5 | 4 | eqeq1d 2825 | . . . . . . 7 ⊢ ({𝑀, 𝑁} = {𝑁} → ((♯‘{𝑁}) = 1 ↔ (♯‘{𝑀, 𝑁}) = 1)) |
6 | 5 | biimpa 479 | . . . . . 6 ⊢ (({𝑀, 𝑁} = {𝑁} ∧ (♯‘{𝑁}) = 1) → (♯‘{𝑀, 𝑁}) = 1) |
7 | id 22 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → (♯‘{𝑀, 𝑁}) = 1) | |
8 | 1ne2 11848 | . . . . . . . . 9 ⊢ 1 ≠ 2 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → 1 ≠ 2) |
10 | 7, 9 | eqnetrd 3085 | . . . . . . 7 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → (♯‘{𝑀, 𝑁}) ≠ 2) |
11 | 10 | neneqd 3023 | . . . . . 6 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → ¬ (♯‘{𝑀, 𝑁}) = 2) |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (({𝑀, 𝑁} = {𝑁} ∧ (♯‘{𝑁}) = 1) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
13 | 12 | expcom 416 | . . . 4 ⊢ ((♯‘{𝑁}) = 1 → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
14 | 2, 13 | syl 17 | . . 3 ⊢ (𝑁 ∈ V → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
15 | snprc 4655 | . . . 4 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅) | |
16 | eqeq2 2835 | . . . . . . 7 ⊢ ({𝑁} = ∅ → ({𝑀, 𝑁} = {𝑁} ↔ {𝑀, 𝑁} = ∅)) | |
17 | 16 | biimpa 479 | . . . . . 6 ⊢ (({𝑁} = ∅ ∧ {𝑀, 𝑁} = {𝑁}) → {𝑀, 𝑁} = ∅) |
18 | hash0 13731 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
19 | fveq2 6672 | . . . . . . . . . 10 ⊢ ({𝑀, 𝑁} = ∅ → (♯‘{𝑀, 𝑁}) = (♯‘∅)) | |
20 | 19 | eqcomd 2829 | . . . . . . . . 9 ⊢ ({𝑀, 𝑁} = ∅ → (♯‘∅) = (♯‘{𝑀, 𝑁})) |
21 | 20 | eqeq1d 2825 | . . . . . . . 8 ⊢ ({𝑀, 𝑁} = ∅ → ((♯‘∅) = 0 ↔ (♯‘{𝑀, 𝑁}) = 0)) |
22 | 21 | biimpa 479 | . . . . . . 7 ⊢ (({𝑀, 𝑁} = ∅ ∧ (♯‘∅) = 0) → (♯‘{𝑀, 𝑁}) = 0) |
23 | id 22 | . . . . . . . . 9 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → (♯‘{𝑀, 𝑁}) = 0) | |
24 | 0ne2 11847 | . . . . . . . . . 10 ⊢ 0 ≠ 2 | |
25 | 24 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → 0 ≠ 2) |
26 | 23, 25 | eqnetrd 3085 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → (♯‘{𝑀, 𝑁}) ≠ 2) |
27 | 26 | neneqd 3023 | . . . . . . 7 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → ¬ (♯‘{𝑀, 𝑁}) = 2) |
28 | 22, 27 | syl 17 | . . . . . 6 ⊢ (({𝑀, 𝑁} = ∅ ∧ (♯‘∅) = 0) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
29 | 17, 18, 28 | sylancl 588 | . . . . 5 ⊢ (({𝑁} = ∅ ∧ {𝑀, 𝑁} = {𝑁}) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
30 | 29 | ex 415 | . . . 4 ⊢ ({𝑁} = ∅ → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
31 | 15, 30 | sylbi 219 | . . 3 ⊢ (¬ 𝑁 ∈ V → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
32 | 14, 31 | pm2.61i 184 | . 2 ⊢ ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2) |
33 | 1, 32 | syl 17 | 1 ⊢ (¬ 𝑀 ∈ V → ¬ (♯‘{𝑀, 𝑁}) = 2) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 {csn 4569 {cpr 4571 ‘cfv 6357 0cc0 10539 1c1 10540 2c2 11695 ♯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-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: hashprb 13761 |
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