Theorem nnawordex 3192

Description: Equivalence for weak ordering of natural numbers.

Assertion

Ref	Expression
nnawordex	⊢ ((A ∈ ω ∧ B ∈ ω) → (A ⊆ B ↔ ∃x ∈ ω (A +o x) = B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem nnawordex

Step	Hyp	Ref	Expression
1		oawordex 3159	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B ↔ ∃x ∈ On (A +o x) = B))
2		nnont 2379	. . . 4 ⊢ (B ∈ ω → B ∈ On)
3	1, 2	sylan2 346	. . 3 ⊢ ((A ∈ On ∧ B ∈ ω) → (A ⊆ B ↔ ∃x ∈ On (A +o x) = B))
4		eleq1 1149	. . . . . . . . . . . . 13 ⊢ ((A +o x) = B → ((A +o x) ∈ ω ↔ B ∈ ω))
5	4	bicomd 399	. . . . . . . . . . . 12 ⊢ ((A +o x) = B → (B ∈ ω ↔ (A +o x) ∈ ω))
6		nnarcl 3174	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ x ∈ On) → ((A +o x) ∈ ω ↔ (A ∈ ω ∧ x ∈ ω)))
7	5, 6	sylan9bbr 419	. . . . . . . . . . 11 ⊢ (((A ∈ On ∧ x ∈ On) ∧ (A +o x) = B) → (B ∈ ω ↔ (A ∈ ω ∧ x ∈ ω)))
8		pm3.27 260	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ x ∈ ω) → x ∈ ω)
9	7, 8	syl6bi 187	. . . . . . . . . 10 ⊢ (((A ∈ On ∧ x ∈ On) ∧ (A +o x) = B) → (B ∈ ω → x ∈ ω))
10	9	exp31 293	. . . . . . . . 9 ⊢ (A ∈ On → (x ∈ On → ((A +o x) = B → (B ∈ ω → x ∈ ω))))
11	10	com23 32	. . . . . . . 8 ⊢ (A ∈ On → ((A +o x) = B → (x ∈ On → (B ∈ ω → x ∈ ω))))
12	11	com24 37	. . . . . . 7 ⊢ (A ∈ On → (B ∈ ω → (x ∈ On → ((A +o x) = B → x ∈ ω))))
13	12	imp4b 283	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ ω) → ((x ∈ On ∧ (A +o x) = B) → x ∈ ω))
14		pm3.27 260	. . . . . . 7 ⊢ ((x ∈ On ∧ (A +o x) = B) → (A +o x) = B)
15	14	a1i 7	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ ω) → ((x ∈ On ∧ (A +o x) = B) → (A +o x) = B))
16	13, 15	jcad 455	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ ω) → ((x ∈ On ∧ (A +o x) = B) → (x ∈ ω ∧ (A +o x) = B)))
17		nnont 2379	. . . . . . 7 ⊢ (x ∈ ω → x ∈ On)
18	17	anim1i 269	. . . . . 6 ⊢ ((x ∈ ω ∧ (A +o x) = B) → (x ∈ On ∧ (A +o x) = B))
19	18	a1i 7	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ ω) → ((x ∈ ω ∧ (A +o x) = B) → (x ∈ On ∧ (A +o x) = B)))
20	16, 19	impbid 397	. . . 4 ⊢ ((A ∈ On ∧ B ∈ ω) → ((x ∈ On ∧ (A +o x) = B) ↔ (x ∈ ω ∧ (A +o x) = B)))
21	20	birexdv2 1222	. . 3 ⊢ ((A ∈ On ∧ B ∈ ω) → (∃x ∈ On (A +o x) = B ↔ ∃x ∈ ω (A +o x) = B))
22	3, 21	bitrd 406	. 2 ⊢ ((A ∈ On ∧ B ∈ ω) → (A ⊆ B ↔ ∃x ∈ ω (A +o x) = B))
23		nnont 2379	. 2 ⊢ (A ∈ ω → A ∈ On)
24	22, 23	sylan 343	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A ⊆ B ↔ ∃x ∈ ω (A +o x) = B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  Oncon0 2199  ωcom 2372  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  unfilem1 3438

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-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  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-int 1966  df-iun 1996  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-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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