Theorem isinf2 37934

Description: The converse of isinf 9221. 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.)

Assertion

Ref	Expression
isinf2	⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin)

Distinct variable group:   𝐴,𝑛,𝑥

Proof of Theorem isinf2

Step	Hyp	Ref	Expression
1		ssdomg 8993	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
2	1	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
3		domen1 9103	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝑛 → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴))
4	3	adantl 486	. . . . . . . 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 458	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑥 ≈ 𝑛 ∧ 𝑥 ⊆ 𝐴) → 𝑛 ≼ 𝐴))
7	6	ancomsd 470	. . . . 5 ⊢ (𝐴 ∈ V → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴))
8	7	exlimdv 1960	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴))
9	8	ralimdv 3185	. . 3 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∀𝑛 ∈ ω 𝑛 ≼ 𝐴))
10		domalom 37933	. . 3 ⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)
11	9, 10	syl6 36	. 2 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin))
12		prcnel 3488	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ Fin)
13	12	a1d 26	. 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin))
14	11, 13	pm2.61i 184	1 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913   class class class wbr 5110  ωcom 7858   ≈ cen 8936   ≼ cdom 8937  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7859  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943

This theorem is referenced by:  ctbssinf  37935