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Mirrors > Home > MPE Home > Th. List > hashnemnf | Structured version Visualization version GIF version |
Description: The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.) |
Ref | Expression |
---|---|
hashnemnf | ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnn0pnf 13705 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | mnfnre 10687 | . . . . . 6 ⊢ -∞ ∉ ℝ | |
3 | df-nel 3127 | . . . . . . 7 ⊢ (-∞ ∉ ℝ ↔ ¬ -∞ ∈ ℝ) | |
4 | nn0re 11909 | . . . . . . . 8 ⊢ (-∞ ∈ ℕ0 → -∞ ∈ ℝ) | |
5 | 4 | con3i 157 | . . . . . . 7 ⊢ (¬ -∞ ∈ ℝ → ¬ -∞ ∈ ℕ0) |
6 | 3, 5 | sylbi 219 | . . . . . 6 ⊢ (-∞ ∉ ℝ → ¬ -∞ ∈ ℕ0) |
7 | 2, 6 | ax-mp 5 | . . . . 5 ⊢ ¬ -∞ ∈ ℕ0 |
8 | eleq1 2903 | . . . . 5 ⊢ ((♯‘𝐴) = -∞ → ((♯‘𝐴) ∈ ℕ0 ↔ -∞ ∈ ℕ0)) | |
9 | 7, 8 | mtbiri 329 | . . . 4 ⊢ ((♯‘𝐴) = -∞ → ¬ (♯‘𝐴) ∈ ℕ0) |
10 | 9 | necon2ai 3048 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ≠ -∞) |
11 | pnfnemnf 10699 | . . . 4 ⊢ +∞ ≠ -∞ | |
12 | neeq1 3081 | . . . 4 ⊢ ((♯‘𝐴) = +∞ → ((♯‘𝐴) ≠ -∞ ↔ +∞ ≠ -∞)) | |
13 | 11, 12 | mpbiri 260 | . . 3 ⊢ ((♯‘𝐴) = +∞ → (♯‘𝐴) ≠ -∞) |
14 | 10, 13 | jaoi 853 | . 2 ⊢ (((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞) → (♯‘𝐴) ≠ -∞) |
15 | 1, 14 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∉ wnel 3126 ‘cfv 6358 ℝcr 10539 +∞cpnf 10675 -∞cmnf 10676 ℕ0cn0 11900 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-hash 13694 |
This theorem is referenced by: hashinfxadd 13749 |
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