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Mirrors > Home > MPE Home > Th. List > hashrabrsn | Structured version Visualization version GIF version |
Description: The size of a restricted class abstraction restricted to a singleton is a nonnegative integer. (Contributed by Alexander van der Vekens, 22-Dec-2017.) |
Ref | Expression |
---|---|
hashrabrsn | ⊢ (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} | |
2 | rabrsn 4686 | . 2 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∈ {𝐴} ∣ 𝜑} → ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})) | |
3 | fveq2 6843 | . . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = (♯‘∅)) | |
4 | hash0 14273 | . . . . 5 ⊢ (♯‘∅) = 0 | |
5 | 0nn0 12433 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
6 | 4, 5 | eqeltri 2830 | . . . 4 ⊢ (♯‘∅) ∈ ℕ0 |
7 | 3, 6 | eqeltrdi 2842 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ → (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0) |
8 | fveq2 6843 | . . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) = (♯‘{𝐴})) | |
9 | hashsng 14275 | . . . . . 6 ⊢ (𝐴 ∈ V → (♯‘{𝐴}) = 1) | |
10 | 1nn0 12434 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
11 | 9, 10 | eqeltrdi 2842 | . . . . 5 ⊢ (𝐴 ∈ V → (♯‘{𝐴}) ∈ ℕ0) |
12 | snprc 4679 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
13 | fveq2 6843 | . . . . . . 7 ⊢ ({𝐴} = ∅ → (♯‘{𝐴}) = (♯‘∅)) | |
14 | 13, 6 | eqeltrdi 2842 | . . . . . 6 ⊢ ({𝐴} = ∅ → (♯‘{𝐴}) ∈ ℕ0) |
15 | 12, 14 | sylbi 216 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (♯‘{𝐴}) ∈ ℕ0) |
16 | 11, 15 | pm2.61i 182 | . . . 4 ⊢ (♯‘{𝐴}) ∈ ℕ0 |
17 | 8, 16 | eqeltrdi 2842 | . . 3 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} → (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0) |
18 | 7, 17 | jaoi 856 | . 2 ⊢ (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) → (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0) |
19 | 1, 2, 18 | mp2b 10 | 1 ⊢ (♯‘{𝑥 ∈ {𝐴} ∣ 𝜑}) ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ∈ wcel 2107 {crab 3406 Vcvv 3444 ∅c0 4283 {csn 4587 ‘cfv 6497 0cc0 11056 1c1 11057 ℕ0cn0 12418 ♯chash 14236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 |
This theorem is referenced by: (None) |
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