Theorem isinfinf 6875

Description: An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)

Assertion

Ref	Expression
isinfinf	⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Distinct variable group:   𝐴,𝑛,𝑥

Proof of Theorem isinfinf

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6727	. . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴)
2	1	adantr 274	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑓 𝑓:ω–1-1→𝐴)
3		vex 2733	. . . . 5 ⊢ 𝑓 ∈ V
4		imaexg 4965	. . . . 5 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑛) ∈ V)
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝑓 “ 𝑛) ∈ V
6		imassrn 4964	. . . . . 6 ⊢ (𝑓 “ 𝑛) ⊆ ran 𝑓
7		simpr 109	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω–1-1→𝐴)
8		f1f 5403	. . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴)
9		frn 5356	. . . . . . 7 ⊢ (𝑓:ω⟶𝐴 → ran 𝑓 ⊆ 𝐴)
10	7, 8, 9	3syl 17	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ⊆ 𝐴)
11	6, 10	sstrid 3158	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ⊆ 𝐴)
12		ordom 4591	. . . . . . . 8 ⊢ Ord ω
13		ordelss 4364	. . . . . . . 8 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
14	12, 13	mpan 422	. . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω)
15	14	ad2antlr 486	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ⊆ ω)
16		simplr 525	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ∈ ω)
17		f1imaeng 6770	. . . . . 6 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ⊆ ω ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ≈ 𝑛)
18	7, 15, 16, 17	syl3anc 1233	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ≈ 𝑛)
19	11, 18	jca 304	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛))
20		sseq1 3170	. . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ 𝑛) ⊆ 𝐴))
21		breq1 3992	. . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓 “ 𝑛) ≈ 𝑛))
22	20, 21	anbi12d 470	. . . . 5 ⊢ (𝑥 = (𝑓 “ 𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛)))
23	22	spcegv 2818	. . . 4 ⊢ ((𝑓 “ 𝑛) ∈ V → (((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
24	5, 19, 23	mpsyl 65	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
25	2, 24	exlimddv 1891	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
26	25	ralrimiva 2543	1 ⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  Vcvv 2730   ⊆ wss 3121   class class class wbr 3989  Ord word 4347  ωcom 4574  ran crn 4612   “ cima 4614  ⟶wf 5194  –1-1→wf1 5195   ≈ cen 6716   ≼ cdom 6717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-dom 6720

This theorem is referenced by: (None)