Theorem infpssr 10379

Description: Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
infpssr	⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)

Proof of Theorem infpssr

Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pssnel 4424	. . 3 ⊢ (𝑋 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋))
2	1	adantr 486	. 2 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋))
3		eldif 3909	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋))
4		pssss 4046	. . . . . 6 ⊢ (𝑋 ⊊ 𝐴 → 𝑋 ⊆ 𝐴)
5		bren 8976	. . . . . . . 8 ⊢ (𝑋 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑋–1-1-onto→𝐴)
6		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑓:𝑋–1-1-onto→𝐴)
7		f1ofo 6830	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋–onto→𝐴)
8		forn 6797	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–onto→𝐴 → ran 𝑓 = 𝐴)
9	6, 7, 8	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → ran 𝑓 = 𝐴)
10		vex 3455	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
11	10	rnex 7920	. . . . . . . . . . . 12 ⊢ ran 𝑓 ∈ V
12	9, 11	eqeltrrdi 2870	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝐴 ∈ V)
13		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑋 ⊆ 𝐴)
14		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑦 ∈ (𝐴 ∖ 𝑋))
15		eqid 2761	. . . . . . . . . . . 12 ⊢ (rec(◡𝑓, 𝑦) ↾ ω) = (rec(◡𝑓, 𝑦) ↾ ω)
16	13, 6, 14, 15	infpssrlem5 10378	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → (𝐴 ∈ V → ω ≼ 𝐴))
17	12, 16	mpd 16	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → ω ≼ 𝐴)
18	17	ex 418	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (𝑓:𝑋–1-1-onto→𝐴 → ω ≼ 𝐴))
19	18	exlimdv 1966	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (∃𝑓 𝑓:𝑋–1-1-onto→𝐴 → ω ≼ 𝐴))
20	5, 19	biimtrid 245	. . . . . . 7 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (𝑋 ≈ 𝐴 → ω ≼ 𝐴))
21	20	ex 418	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → (𝑋 ⊆ 𝐴 → (𝑋 ≈ 𝐴 → ω ≼ 𝐴)))
22	4, 21	syl5 35	. . . . 5 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → (𝑋 ⊊ 𝐴 → (𝑋 ≈ 𝐴 → ω ≼ 𝐴)))
23	22	impd 416	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴))
24	3, 23	sylbir 238	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴))
25	24	exlimiv 1963	. 2 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴))
26	2, 25	mpcom 39	1 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899   ⊊ wpss 3900   class class class wbr 5103  ^◡ccnv 5650  ran crn 5652   ↾ cres 5653  –onto→wfo 6535  –1-1-onto→wf1o 6536  ωcom 7875  reccrdg 8410   ≈ cen 8963   ≼ cdom 8964

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-en 8967  df-dom 8968

This theorem is used by:  isfin4-2  10385