Theorem infpssr 10302

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 4470	. . 3 ⊢ (𝑋 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋))
2	1	adantr 481	. 2 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋))
3		eldif 3958	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋))
4		pssss 4095	. . . . . 6 ⊢ (𝑋 ⊊ 𝐴 → 𝑋 ⊆ 𝐴)
5		bren 8948	. . . . . . . 8 ⊢ (𝑋 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑋–1-1-onto→𝐴)
6		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑓:𝑋–1-1-onto→𝐴)
7		f1ofo 6840	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋–onto→𝐴)
8		forn 6808	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–onto→𝐴 → ran 𝑓 = 𝐴)
9	6, 7, 8	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → ran 𝑓 = 𝐴)
10		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
11	10	rnex 7902	. . . . . . . . . . . 12 ⊢ ran 𝑓 ∈ V
12	9, 11	eqeltrrdi 2842	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝐴 ∈ V)
13		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑋 ⊆ 𝐴)
14		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑦 ∈ (𝐴 ∖ 𝑋))
15		eqid 2732	. . . . . . . . . . . 12 ⊢ (rec(◡𝑓, 𝑦) ↾ ω) = (rec(◡𝑓, 𝑦) ↾ ω)
16	13, 6, 14, 15	infpssrlem5 10301	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → (𝐴 ∈ V → ω ≼ 𝐴))
17	12, 16	mpd 15	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → ω ≼ 𝐴)
18	17	ex 413	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (𝑓:𝑋–1-1-onto→𝐴 → ω ≼ 𝐴))
19	18	exlimdv 1936	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (∃𝑓 𝑓:𝑋–1-1-onto→𝐴 → ω ≼ 𝐴))
20	5, 19	biimtrid 241	. . . . . . 7 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (𝑋 ≈ 𝐴 → ω ≼ 𝐴))
21	20	ex 413	. . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → (𝑋 ⊆ 𝐴 → (𝑋 ≈ 𝐴 → ω ≼ 𝐴)))
22	4, 21	syl5 34	. . . . 5 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → (𝑋 ⊊ 𝐴 → (𝑋 ≈ 𝐴 → ω ≼ 𝐴)))
23	22	impd 411	. . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴))
24	3, 23	sylbir 234	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴))
25	24	exlimiv 1933	. 2 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴))
26	2, 25	mpcom 38	1 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948   ⊊ wpss 3949   class class class wbr 5148  ^◡ccnv 5675  ran crn 5677   ↾ cres 5678  –onto→wfo 6541  –1-1-onto→wf1o 6542  ωcom 7854  reccrdg 8408   ≈ cen 8935   ≼ cdom 8936

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-en 8939  df-dom 8940

This theorem is referenced by:  isfin4-2  10308