Theorem isinf2 37439

Description: The converse of isinf 9144. 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 8917	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
3		domen1 9027	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝑛 → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴))
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ≼ 𝐴 ↔ 𝑛 ≼ 𝐴))
5	2, 4	sylibd 239	. . . . . . 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 1934	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴))
9	8	ralimdv 3146	. . 3 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∀𝑛 ∈ ω 𝑛 ≼ 𝐴))
10		domalom 37438	. . 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 3462	. . 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 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897   class class class wbr 5086  ωcom 7791   ≈ cen 8861   ≼ cdom 8862  Fincfn 8864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-om 7792  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868

This theorem is referenced by:  ctbssinf  37440