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Theorem nnawordex 6548
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) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnawordex
StepHypRef Expression
1 nntri3or 6512 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
213adant3 1019 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3 nnaordex 6547 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
4 simpr 110 . . . . . . . 8 ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
54reximi 2587 . . . . . . 7 (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
63, 5biimtrdi 163 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
763adant3 1019 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
8 nna0 6493 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
983ad2ant1 1020 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
10 eqeq2 2199 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 +o ∅) = 𝐴 ↔ (𝐴 +o ∅) = 𝐵))
119, 10syl5ibcom 155 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 = 𝐵 → (𝐴 +o ∅) = 𝐵))
12 peano1 4608 . . . . . . 7 ∅ ∈ ω
13 oveq2 5899 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
1413eqeq1d 2198 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o ∅) = 𝐵))
1514rspcev 2856 . . . . . . 7 ((∅ ∈ ω ∧ (𝐴 +o ∅) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
1612, 15mpan 424 . . . . . 6 ((𝐴 +o ∅) = 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
1711, 16syl6 33 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 = 𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
18 nntri1 6515 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1918biimp3a 1356 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
2019pm2.21d 620 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵𝐴 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
217, 17, 203jaod 1315 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
222, 21mpd 13 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
23223expia 1207 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
24 nnaword1 6532 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
25 sseq2 3194 . . . . 5 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
2624, 25syl5ibcom 155 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2726rexlimdva 2607 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2827adantr 276 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2923, 28impbid 129 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3o 979  w3a 980   = wceq 1364  wcel 2160  wrex 2469  wss 3144  c0 3437  ωcom 4604  (class class class)co 5891   +o coa 6432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-oadd 6439
This theorem is referenced by:  prarloclemn  7517
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