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Mirrors > Home > MPE Home > Th. List > Mathboxes > isinf2 | Structured version Visualization version GIF version |
Description: The converse of isinf 8867. Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by ML, 14-Dec-2020.) |
Ref | Expression |
---|---|
isinf2 | ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdomg 8652 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴)) | |
2 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴)) |
3 | domen1 8766 | . . . . . . . . 9 ⊢ (𝑥 ≈ 𝑛 → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴)) | |
4 | 3 | adantl 485 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴)) |
5 | 2, 4 | sylibd 242 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑛 ≼ 𝐴)) |
6 | 5 | expimpd 457 | . . . . . 6 ⊢ (𝐴 ∈ V → ((𝑥 ≈ 𝑛 ∧ 𝑥 ⊆ 𝐴) → 𝑛 ≼ 𝐴)) |
7 | 6 | ancomsd 469 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴)) |
8 | 7 | exlimdv 1941 | . . . 4 ⊢ (𝐴 ∈ V → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴)) |
9 | 8 | ralimdv 3091 | . . 3 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∀𝑛 ∈ ω 𝑛 ≼ 𝐴)) |
10 | domalom 35261 | . . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin) | |
11 | 9, 10 | syl6 35 | . 2 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin)) |
12 | prcnel 3421 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ Fin) | |
13 | 12 | a1d 25 | . 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin)) |
14 | 11, 13 | pm2.61i 185 | 1 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1787 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 class class class wbr 5039 ωcom 7622 ≈ cen 8601 ≼ cdom 8602 Fincfn 8604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 |
This theorem is referenced by: ctbssinf 35263 |
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