Theorem omsdomnn 3424

Description: Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. Here we use A ≼ ω ∧ ¬ ω ≈ A instead of A ≺ ω because, due to a peculiarity ultimately caused our ordered pair definition, we would need the axiom of infinity (which we have avoided up to now) in order to prove the latter.

Assertion

Ref	Expression
omsdomnn	⊢ (A ∈ ω → (A ≼ ω ∧ ¬ ω ≈ A))

Proof of Theorem omsdomnn

Step	Hyp	Ref	Expression
1		ordom 2382	. . . 4 ⊢ Ord ω
2		ordelss 2215	. . . 4 ⊢ ((Ord ω ∧ A ∈ ω) → A ⊆ ω)
3	1, 2	mpan 518	. . 3 ⊢ (A ∈ ω → A ⊆ ω)
4		ssdomg 3311	. . 3 ⊢ (A ∈ ω → (A ⊆ ω → A ≼ ω))
5	3, 4	mpd 46	. 2 ⊢ (A ∈ ω → A ≼ ω)
6		breq2 2066	. . . 4 ⊢ (x = A → (ω ≈ x ↔ ω ≈ A))
7	6	negbid 463	. . 3 ⊢ (x = A → (¬ ω ≈ x ↔ ¬ ω ≈ A))
8		ominf 3423	. . . 4 ⊢ ¬ ∃x ∈ ω ω ≈ x
9		ralnex 1209	. . . 4 ⊢ (∀x ∈ ω ¬ ω ≈ x ↔ ¬ ∃x ∈ ω ω ≈ x)
10	8, 9	mpbir 165	. . 3 ⊢ ∀x ∈ ω ¬ ω ≈ x
11	7, 10	vtoclri 1393	. 2 ⊢ (A ∈ ω → ¬ ω ≈ A)
12	5, 11	jca 236	1 ⊢ (A ∈ ω → (A ≼ ω ∧ ¬ ω ≈ A))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487   class class class wbr 2054  Ord word 2198  ωcom 2372   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  isfinite1 3425  infsdomnn 3426  nnsdom 3481  infunabs 4946  infcdaabs 4947

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276

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