Theorem prcinf 35086

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.)

Assertion

Ref	Expression
prcinf	⊢ (¬ 𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Distinct variable group:   𝐴,𝑛,𝑥

Proof of Theorem prcinf

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))

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)