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Theorem nnawordex 6037
Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnawordex ((A 𝜔 B 𝜔) → (ABx 𝜔 (A +𝑜 x) = B))
Distinct variable groups:   x,A   x,B

Proof of Theorem nnawordex
StepHypRef Expression
1 nntri3or 6011 . . . . 5 ((A 𝜔 B 𝜔) → (A B A = B B A))
213adant3 923 . . . 4 ((A 𝜔 B 𝜔 AB) → (A B A = B B A))
3 nnaordex 6036 . . . . . . 7 ((A 𝜔 B 𝜔) → (A Bx 𝜔 (∅ x (A +𝑜 x) = B)))
4 simpr 103 . . . . . . . 8 ((∅ x (A +𝑜 x) = B) → (A +𝑜 x) = B)
54reximi 2410 . . . . . . 7 (x 𝜔 (∅ x (A +𝑜 x) = B) → x 𝜔 (A +𝑜 x) = B)
63, 5syl6bi 152 . . . . . 6 ((A 𝜔 B 𝜔) → (A Bx 𝜔 (A +𝑜 x) = B))
763adant3 923 . . . . 5 ((A 𝜔 B 𝜔 AB) → (A Bx 𝜔 (A +𝑜 x) = B))
8 nna0 5992 . . . . . . . 8 (A 𝜔 → (A +𝑜 ∅) = A)
983ad2ant1 924 . . . . . . 7 ((A 𝜔 B 𝜔 AB) → (A +𝑜 ∅) = A)
10 eqeq2 2046 . . . . . . 7 (A = B → ((A +𝑜 ∅) = A ↔ (A +𝑜 ∅) = B))
119, 10syl5ibcom 144 . . . . . 6 ((A 𝜔 B 𝜔 AB) → (A = B → (A +𝑜 ∅) = B))
12 peano1 4260 . . . . . . 7 𝜔
13 oveq2 5463 . . . . . . . . 9 (x = ∅ → (A +𝑜 x) = (A +𝑜 ∅))
1413eqeq1d 2045 . . . . . . . 8 (x = ∅ → ((A +𝑜 x) = B ↔ (A +𝑜 ∅) = B))
1514rspcev 2650 . . . . . . 7 ((∅ 𝜔 (A +𝑜 ∅) = B) → x 𝜔 (A +𝑜 x) = B)
1612, 15mpan 400 . . . . . 6 ((A +𝑜 ∅) = Bx 𝜔 (A +𝑜 x) = B)
1711, 16syl6 29 . . . . 5 ((A 𝜔 B 𝜔 AB) → (A = Bx 𝜔 (A +𝑜 x) = B))
18 nntri1 6013 . . . . . . 7 ((A 𝜔 B 𝜔) → (AB ↔ ¬ B A))
1918biimp3a 1234 . . . . . 6 ((A 𝜔 B 𝜔 AB) → ¬ B A)
2019pm2.21d 549 . . . . 5 ((A 𝜔 B 𝜔 AB) → (B Ax 𝜔 (A +𝑜 x) = B))
217, 17, 203jaod 1198 . . . 4 ((A 𝜔 B 𝜔 AB) → ((A B A = B B A) → x 𝜔 (A +𝑜 x) = B))
222, 21mpd 13 . . 3 ((A 𝜔 B 𝜔 AB) → x 𝜔 (A +𝑜 x) = B)
23223expia 1105 . 2 ((A 𝜔 B 𝜔) → (ABx 𝜔 (A +𝑜 x) = B))
24 nnaword1 6022 . . . . 5 ((A 𝜔 x 𝜔) → A ⊆ (A +𝑜 x))
25 sseq2 2961 . . . . 5 ((A +𝑜 x) = B → (A ⊆ (A +𝑜 x) ↔ AB))
2624, 25syl5ibcom 144 . . . 4 ((A 𝜔 x 𝜔) → ((A +𝑜 x) = BAB))
2726rexlimdva 2427 . . 3 (A 𝜔 → (x 𝜔 (A +𝑜 x) = BAB))
2827adantr 261 . 2 ((A 𝜔 B 𝜔) → (x 𝜔 (A +𝑜 x) = BAB))
2923, 28impbid 120 1 ((A 𝜔 B 𝜔) → (ABx 𝜔 (A +𝑜 x) = B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3o 883   w3a 884   = wceq 1242   wcel 1390  wrex 2301  wss 2911  c0 3218  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944
This theorem is referenced by:  prarloclemn  6481
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