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Theorem nnawordex 6008
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 5983 . . . . 5 ((A 𝜔 B 𝜔) → (A B A = B B A))
213adant3 910 . . . 4 ((A 𝜔 B 𝜔 AB) → (A B A = B B A))
3 nnaordex 6007 . . . . . . 7 ((A 𝜔 B 𝜔) → (A Bx 𝜔 (∅ x (A +𝑜 x) = B)))
4 ax-ia2 100 . . . . . . . 8 ((∅ x (A +𝑜 x) = B) → (A +𝑜 x) = B)
54reximi 2390 . . . . . . 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 910 . . . . 5 ((A 𝜔 B 𝜔 AB) → (A Bx 𝜔 (A +𝑜 x) = B))
8 nna0 5964 . . . . . . . 8 (A 𝜔 → (A +𝑜 ∅) = A)
983ad2ant1 911 . . . . . . 7 ((A 𝜔 B 𝜔 AB) → (A +𝑜 ∅) = A)
10 eqeq2 2027 . . . . . . 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 4240 . . . . . . 7 𝜔
13 oveq2 5440 . . . . . . . . 9 (x = ∅ → (A +𝑜 x) = (A +𝑜 ∅))
1413eqeq1d 2026 . . . . . . . 8 (x = ∅ → ((A +𝑜 x) = B ↔ (A +𝑜 ∅) = B))
1514rspcev 2629 . . . . . . 7 ((∅ 𝜔 (A +𝑜 ∅) = B) → x 𝜔 (A +𝑜 x) = B)
1612, 15mpan 402 . . . . . 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 5985 . . . . . . 7 ((A 𝜔 B 𝜔) → (AB ↔ ¬ B A))
1918biimp3a 1218 . . . . . 6 ((A 𝜔 B 𝜔 AB) → ¬ B A)
2019pm2.21d 537 . . . . 5 ((A 𝜔 B 𝜔 AB) → (B Ax 𝜔 (A +𝑜 x) = B))
217, 17, 203jaod 1183 . . . 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 1090 . 2 ((A 𝜔 B 𝜔) → (ABx 𝜔 (A +𝑜 x) = B))
24 nnaword1 5993 . . . . 5 ((A 𝜔 x 𝜔) → A ⊆ (A +𝑜 x))
25 sseq2 2940 . . . . 5 ((A +𝑜 x) = B → (A ⊆ (A +𝑜 x) ↔ AB))
2624, 25syl5ibcom 144 . . . 4 ((A 𝜔 x 𝜔) → ((A +𝑜 x) = BAB))
2726rexlimdva 2407 . . 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 870   w3a 871   = wceq 1226   wcel 1370  wrex 2281  wss 2890  c0 3197  𝜔com 4236  (class class class)co 5432   +𝑜 coa 5909
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-1o 5912  df-oadd 5916
This theorem is referenced by:  prarloclemn  6347
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