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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prcinf | Structured version Visualization version GIF version | ||
| Description: Any proper class is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. This proof holds regardless of whether the Axiom of Infinity is accepted or negated. (Contributed by BTernaryTau, 22-Jun-2025.) |
| Ref | Expression |
|---|---|
| prcinf | ⊢ (¬ 𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3500 | . 2 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V) | |
| 2 | isinf 9292 | . 2 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) | |
| 3 | 1, 2 | nsyl5 159 | 1 ⊢ (¬ 𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ∀wral 3060 Vcvv 3479 ⊆ wss 3950 class class class wbr 5141 ωcom 7883 ≈ cen 8978 Fincfn 8981 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-om 7884 df-en 8982 df-fin 8985 |
| This theorem is referenced by: (None) |
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