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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 4770 | . 2 ⊢ (¬ 𝑀 ∈ V → {𝑀, 𝑁} = {𝑁}) | |
2 | hashsng 14405 | . . . 4 ⊢ (𝑁 ∈ V → (♯‘{𝑁}) = 1) | |
3 | fveq2 6907 | . . . . . . . . 9 ⊢ ({𝑀, 𝑁} = {𝑁} → (♯‘{𝑀, 𝑁}) = (♯‘{𝑁})) | |
4 | 3 | eqcomd 2741 | . . . . . . . 8 ⊢ ({𝑀, 𝑁} = {𝑁} → (♯‘{𝑁}) = (♯‘{𝑀, 𝑁})) |
5 | 4 | eqeq1d 2737 | . . . . . . 7 ⊢ ({𝑀, 𝑁} = {𝑁} → ((♯‘{𝑁}) = 1 ↔ (♯‘{𝑀, 𝑁}) = 1)) |
6 | 5 | biimpa 476 | . . . . . 6 ⊢ (({𝑀, 𝑁} = {𝑁} ∧ (♯‘{𝑁}) = 1) → (♯‘{𝑀, 𝑁}) = 1) |
7 | id 22 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → (♯‘{𝑀, 𝑁}) = 1) | |
8 | 1ne2 12472 | . . . . . . . . 9 ⊢ 1 ≠ 2 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → 1 ≠ 2) |
10 | 7, 9 | eqnetrd 3006 | . . . . . . 7 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → (♯‘{𝑀, 𝑁}) ≠ 2) |
11 | 10 | neneqd 2943 | . . . . . 6 ⊢ ((♯‘{𝑀, 𝑁}) = 1 → ¬ (♯‘{𝑀, 𝑁}) = 2) |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (({𝑀, 𝑁} = {𝑁} ∧ (♯‘{𝑁}) = 1) → ¬ (♯‘{𝑀, 𝑁}) = 2) |
13 | 12 | expcom 413 | . . . 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 4722 | . . . 4 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅) | |
16 | eqeq2 2747 | . . . . . . 7 ⊢ ({𝑁} = ∅ → ({𝑀, 𝑁} = {𝑁} ↔ {𝑀, 𝑁} = ∅)) | |
17 | 16 | biimpa 476 | . . . . . 6 ⊢ (({𝑁} = ∅ ∧ {𝑀, 𝑁} = {𝑁}) → {𝑀, 𝑁} = ∅) |
18 | hash0 14403 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
19 | fveq2 6907 | . . . . . . . . . 10 ⊢ ({𝑀, 𝑁} = ∅ → (♯‘{𝑀, 𝑁}) = (♯‘∅)) | |
20 | 19 | eqcomd 2741 | . . . . . . . . 9 ⊢ ({𝑀, 𝑁} = ∅ → (♯‘∅) = (♯‘{𝑀, 𝑁})) |
21 | 20 | eqeq1d 2737 | . . . . . . . 8 ⊢ ({𝑀, 𝑁} = ∅ → ((♯‘∅) = 0 ↔ (♯‘{𝑀, 𝑁}) = 0)) |
22 | 21 | biimpa 476 | . . . . . . 7 ⊢ (({𝑀, 𝑁} = ∅ ∧ (♯‘∅) = 0) → (♯‘{𝑀, 𝑁}) = 0) |
23 | id 22 | . . . . . . . . 9 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → (♯‘{𝑀, 𝑁}) = 0) | |
24 | 0ne2 12471 | . . . . . . . . . 10 ⊢ 0 ≠ 2 | |
25 | 24 | a1i 11 | . . . . . . . . 9 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → 0 ≠ 2) |
26 | 23, 25 | eqnetrd 3006 | . . . . . . . 8 ⊢ ((♯‘{𝑀, 𝑁}) = 0 → (♯‘{𝑀, 𝑁}) ≠ 2) |
27 | 26 | neneqd 2943 | . . . . . . 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 412 | . . . 4 ⊢ ({𝑁} = ∅ → ({𝑀, 𝑁} = {𝑁} → ¬ (♯‘{𝑀, 𝑁}) = 2)) |
31 | 15, 30 | sylbi 217 | . . 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {csn 4631 {cpr 4633 ‘cfv 6563 0cc0 11153 1c1 11154 2c2 12319 ♯chash 14366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 |
This theorem is referenced by: hashprb 14433 |
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