![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hashrabsn01 | Structured version Visualization version GIF version |
Description: The size of a restricted class abstraction restricted to a singleton is either 0 or 1. (Contributed by Alexander van der Vekens, 3-Sep-2018.) |
Ref | Expression |
---|---|
hashrabsn01 | ⊢ ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} | |
2 | rabrsn 4620 | . 2 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} → ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})) | |
3 | fveqeq2 6654 | . . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 ↔ (♯‘∅) = 𝑁)) | |
4 | eqcom 2805 | . . . . . . 7 ⊢ ((♯‘∅) = 𝑁 ↔ 𝑁 = (♯‘∅)) | |
5 | 4 | biimpi 219 | . . . . . 6 ⊢ ((♯‘∅) = 𝑁 → 𝑁 = (♯‘∅)) |
6 | hash0 13724 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
7 | 5, 6 | eqtrdi 2849 | . . . . 5 ⊢ ((♯‘∅) = 𝑁 → 𝑁 = 0) |
8 | 7 | orcd 870 | . . . 4 ⊢ ((♯‘∅) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1)) |
9 | 3, 8 | syl6bi 256 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))) |
10 | fveqeq2 6654 | . . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 ↔ (♯‘{𝐴}) = 𝑁)) | |
11 | eqcom 2805 | . . . . . . . . 9 ⊢ ((♯‘{𝐴}) = 𝑁 ↔ 𝑁 = (♯‘{𝐴})) | |
12 | 11 | biimpi 219 | . . . . . . . 8 ⊢ ((♯‘{𝐴}) = 𝑁 → 𝑁 = (♯‘{𝐴})) |
13 | hashsng 13726 | . . . . . . . 8 ⊢ (𝐴 ∈ V → (♯‘{𝐴}) = 1) | |
14 | 12, 13 | sylan9eqr 2855 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ (♯‘{𝐴}) = 𝑁) → 𝑁 = 1) |
15 | 14 | olcd 871 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ (♯‘{𝐴}) = 𝑁) → (𝑁 = 0 ∨ 𝑁 = 1)) |
16 | 15 | ex 416 | . . . . 5 ⊢ (𝐴 ∈ V → ((♯‘{𝐴}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))) |
17 | snprc 4613 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
18 | fveqeq2 6654 | . . . . . . 7 ⊢ ({𝐴} = ∅ → ((♯‘{𝐴}) = 𝑁 ↔ (♯‘∅) = 𝑁)) | |
19 | 18, 8 | syl6bi 256 | . . . . . 6 ⊢ ({𝐴} = ∅ → ((♯‘{𝐴}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))) |
20 | 17, 19 | sylbi 220 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ((♯‘{𝐴}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))) |
21 | 16, 20 | pm2.61i 185 | . . . 4 ⊢ ((♯‘{𝐴}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1)) |
22 | 10, 21 | syl6bi 256 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))) |
23 | 9, 22 | jaoi 854 | . 2 ⊢ (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1))) |
24 | 1, 2, 23 | mp2b 10 | 1 ⊢ ((♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∅c0 4243 {csn 4525 ‘cfv 6324 0cc0 10526 1c1 10527 ♯chash 13686 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |