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Theorem nnawordex 6424
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 6389 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
213adant3 1001 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3 nnaordex 6423 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
4 simpr 109 . . . . . . . 8 ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
54reximi 2529 . . . . . . 7 (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
63, 5syl6bi 162 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
763adant3 1001 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
8 nna0 6370 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
983ad2ant1 1002 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
10 eqeq2 2149 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 +o ∅) = 𝐴 ↔ (𝐴 +o ∅) = 𝐵))
119, 10syl5ibcom 154 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 = 𝐵 → (𝐴 +o ∅) = 𝐵))
12 peano1 4508 . . . . . . 7 ∅ ∈ ω
13 oveq2 5782 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
1413eqeq1d 2148 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o ∅) = 𝐵))
1514rspcev 2789 . . . . . . 7 ((∅ ∈ ω ∧ (𝐴 +o ∅) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
1612, 15mpan 420 . . . . . 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 6392 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1918biimp3a 1323 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
2019pm2.21d 608 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵𝐴 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
217, 17, 203jaod 1282 . . . 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 1183 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
24 nnaword1 6409 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
25 sseq2 3121 . . . . 5 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
2624, 25syl5ibcom 154 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2726rexlimdva 2549 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2827adantr 274 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2923, 28impbid 128 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 103  wb 104  w3o 961  w3a 962   = wceq 1331  wcel 1480  wrex 2417  wss 3071  c0 3363  ωcom 4504  (class class class)co 5774   +o coa 6310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317
This theorem is referenced by:  prarloclemn  7319
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