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Mirrors > Home > MPE Home > Th. List > hashnnn0genn0 | Structured version Visualization version GIF version |
Description: If the size of a set is not a nonnegative integer, it is greater than or equal to any nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.) |
Ref | Expression |
---|---|
hashnnn0genn0 | ⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (♯‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3047 | . . . 4 ⊢ ((♯‘𝑀) ∉ ℕ0 ↔ ¬ (♯‘𝑀) ∈ ℕ0) | |
2 | pm2.21 123 | . . . 4 ⊢ (¬ (♯‘𝑀) ∈ ℕ0 → ((♯‘𝑀) ∈ ℕ0 → 𝑁 ≤ (♯‘𝑀))) | |
3 | 1, 2 | sylbi 216 | . . 3 ⊢ ((♯‘𝑀) ∉ ℕ0 → ((♯‘𝑀) ∈ ℕ0 → 𝑁 ≤ (♯‘𝑀))) |
4 | 3 | 3ad2ant2 1135 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑀) ∈ ℕ0 → 𝑁 ≤ (♯‘𝑀))) |
5 | nn0re 12427 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 5 | ltpnfd 13047 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 < +∞) |
7 | 5 | rexrd 11210 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ*) |
8 | pnfxr 11214 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
9 | xrltle 13074 | . . . . . 6 ⊢ ((𝑁 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑁 < +∞ → 𝑁 ≤ +∞)) | |
10 | 7, 8, 9 | sylancl 587 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < +∞ → 𝑁 ≤ +∞)) |
11 | 6, 10 | mpd 15 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ +∞) |
12 | breq2 5110 | . . . 4 ⊢ ((♯‘𝑀) = +∞ → (𝑁 ≤ (♯‘𝑀) ↔ 𝑁 ≤ +∞)) | |
13 | 11, 12 | syl5ibrcom 247 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝑀) = +∞ → 𝑁 ≤ (♯‘𝑀))) |
14 | 13 | 3ad2ant3 1136 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑀) = +∞ → 𝑁 ≤ (♯‘𝑀))) |
15 | hashnn0pnf 14248 | . . 3 ⊢ (𝑀 ∈ 𝑉 → ((♯‘𝑀) ∈ ℕ0 ∨ (♯‘𝑀) = +∞)) | |
16 | 15 | 3ad2ant1 1134 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑀) ∈ ℕ0 ∨ (♯‘𝑀) = +∞)) |
17 | 4, 14, 16 | mpjaod 859 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ (♯‘𝑀) ∉ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (♯‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∉ wnel 3046 class class class wbr 5106 ‘cfv 6497 +∞cpnf 11191 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 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-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-xnn0 12491 df-z 12505 df-uz 12769 df-hash 14237 |
This theorem is referenced by: (None) |
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