Theorem isinf2 37657

Description: The converse of isinf 9177. 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 8949	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝑥 ≈ 𝑛) → (𝑥 ⊆ 𝐴 → 𝑥 ≼ 𝐴))
3		domen1 9059	. . . . . . . . 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 1935	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → 𝑛 ≼ 𝐴))
9	8	ralimdv 3152	. . 3 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∀𝑛 ∈ ω 𝑛 ≼ 𝐴))
10		domalom 37656	. . 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 3468	. . 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 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903   class class class wbr 5100  ωcom 7818   ≈ cen 8892   ≼ cdom 8893  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899

This theorem is referenced by:  ctbssinf  37658