Theorem unbnn 9208

Description: Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 9580 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)

Assertion

Ref	Expression
unbnn	⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem unbnn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssdomg 8949	. . . 4 ⊢ (ω ∈ V → (𝐴 ⊆ ω → 𝐴 ≼ ω))
2	1	imp 406	. . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ≼ ω)
3	2	3adant3 1133	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≼ ω)
4		simp1 1137	. . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ∈ V)
5		ssexg 5270	. . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ω ∈ V) → 𝐴 ∈ V)
6	5	ancoms 458	. . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω) → 𝐴 ∈ V)
7	6	3adant3 1133	. . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
8		eqid 2737	. . . . 5 ⊢ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω) = (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω)
9	8	unblem4 9207	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴)
10	9	3adant1 1131	. . 3 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴)
11		f1dom2g 8918	. . 3 ⊢ ((ω ∈ V ∧ 𝐴 ∈ V ∧ (rec((𝑧 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑧)), ∩ 𝐴) ↾ ω):ω–1-1→𝐴) → ω ≼ 𝐴)
12	4, 7, 10, 11	syl3anc 1374	. 2 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ω ≼ 𝐴)
13		sbth 9037	. 2 ⊢ ((𝐴 ≼ ω ∧ ω ≼ 𝐴) → 𝐴 ≈ ω)
14	3, 12, 13	syl2anc 585	1 ⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∩ cint 4904   class class class wbr 5100   ↦ cmpt 5181   ↾ cres 5634  suc csuc 6327  –1-1→wf1 6497  ωcom 7818  reccrdg 8350   ≈ cen 8892   ≼ cdom 8893

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-int 4905  df-iun 4950  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-pred 6267  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-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-en 8896  df-dom 8897

This theorem is referenced by:  unbnn2  9209  isfinite2  9210  unbnn3  9580