Theorem isinf2 35503

Description: The converse of isinf 8965. 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 8741	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
3		domen1 8855	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝑛 → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴))
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴))
5	2, 4	sylibd 238	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑛 ≼ 𝐴))
6	5	expimpd 453	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑥 ≈ 𝑛 ∧ 𝑥 ⊆ 𝐴) → 𝑛 ≼ 𝐴))
7	6	ancomsd 465	. . . . 5 ⊢ (𝐴 ∈ V → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴))
8	7	exlimdv 1937	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴))
9	8	ralimdv 3103	. . 3 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∀𝑛 ∈ ω 𝑛 ≼ 𝐴))
10		domalom 35502	. . 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 3445	. . 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 182	1 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883   class class class wbr 5070  ωcom 7687   ≈ cen 8688   ≼ cdom 8689  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695

This theorem is referenced by:  ctbssinf  35504