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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 4726 | . 2 ⊢ (¬ 𝑀 ∈ V → {𝑀, 𝑁} = {𝑁}) | |
2 | hashsng 14269 | . . . 4 ⊢ (𝑁 ∈ V → (♯‘{𝑁}) = 1) | |
3 | fveq2 6842 | . . . . . . . . 9 ⊢ ({𝑀, 𝑁} = {𝑁} → (♯‘{𝑀, 𝑁}) = (♯‘{𝑁})) | |
4 | 3 | eqcomd 2742 | . . . . . . . 8 ⊢ ({𝑀, 𝑁} = {𝑁} → (♯‘{𝑁}) = (♯‘{𝑀, 𝑁})) |
5 | 4 | eqeq1d 2738 | . . . . . . 7 ⊢ ({𝑀, 𝑁} = {𝑁} → ((♯‘{𝑁}) = 1 ↔ (♯‘{𝑀, 𝑁}) = 1)) |
6 | 5 | biimpa 477 | . . . . . 6 ⊢ (({𝑀, 𝑁} = {𝑁} ∧ (♯‘{𝑁}) = 1) → (♯‘{𝑀, 𝑁}) = 1) |
7 | id 22 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → (♯‘{𝑀, 𝑁}) = 1) | |
8 | 1ne2 12361 | . . . . . . . . 9 ⊢ 1 ≠ 2 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → 1 ≠ 2) |
10 | 7, 9 | eqnetrd 3011 | . . . . . . 7 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → (♯‘{𝑀, 𝑁}) ≠ 2) |
11 | 10 | neneqd 2948 | . . . . . 6 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → ¬ (♯‘{𝑀, 𝑁}) = 2) |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (({𝑀, 𝑁} = {𝑁} ∧ (♯‘{𝑁}) = 1) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
13 | 12 | expcom 414 | . . . 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 4678 | . . . 4 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅) | |
16 | eqeq2 2748 | . . . . . . 7 ⊢ ({𝑁} = ∅ → ({𝑀, 𝑁} = {𝑁} ↔ {𝑀, 𝑁} = ∅)) | |
17 | 16 | biimpa 477 | . . . . . 6 ⊢ (({𝑁} = ∅ ∧ {𝑀, 𝑁} = {𝑁}) → {𝑀, 𝑁} = ∅) |
18 | hash0 14267 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
19 | fveq2 6842 | . . . . . . . . . 10 ⊢ ({𝑀, 𝑁} = ∅ → (♯‘{𝑀, 𝑁}) = (♯‘∅)) | |
20 | 19 | eqcomd 2742 | . . . . . . . . 9 ⊢ ({𝑀, 𝑁} = ∅ → (♯‘∅) = (♯‘{𝑀, 𝑁})) |
21 | 20 | eqeq1d 2738 | . . . . . . . 8 ⊢ ({𝑀, 𝑁} = ∅ → ((♯‘∅) = 0 ↔ (♯‘{𝑀, 𝑁}) = 0)) |
22 | 21 | biimpa 477 | . . . . . . 7 ⊢ (({𝑀, 𝑁} = ∅ ∧ (♯‘∅) = 0) → (♯‘{𝑀, 𝑁}) = 0) |
23 | id 22 | . . . . . . . . 9 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → (♯‘{𝑀, 𝑁}) = 0) | |
24 | 0ne2 12360 | . . . . . . . . . 10 ⊢ 0 ≠ 2 | |
25 | 24 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → 0 ≠ 2) |
26 | 23, 25 | eqnetrd 3011 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → (♯‘{𝑀, 𝑁}) ≠ 2) |
27 | 26 | neneqd 2948 | . . . . . . 7 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → ¬ (♯‘{𝑀, 𝑁}) = 2) |
28 | 22, 27 | syl 17 | . . . . . 6 ⊢ (({𝑀, 𝑁} = ∅ ∧ (♯‘∅) = 0) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
29 | 17, 18, 28 | sylancl 586 | . . . . 5 ⊢ (({𝑁} = ∅ ∧ {𝑀, 𝑁} = {𝑁}) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
30 | 29 | ex 413 | . . . 4 ⊢ ({𝑁} = ∅ → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
31 | 15, 30 | sylbi 216 | . . 3 ⊢ (¬ 𝑁 ∈ V → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
32 | 14, 31 | pm2.61i 182 | . 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ∅c0 4282 {csn 4586 {cpr 4588 ‘cfv 6496 0cc0 11051 1c1 11052 2c2 12208 ♯chash 14230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231 |
This theorem is referenced by: hashprb 14297 |
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