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Theorem nnawordex 6532
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 6496 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
213adant3 1017 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3 nnaordex 6531 . . . . . . 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 2574 . . . . . . 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 1017 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
8 nna0 6477 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
983ad2ant1 1018 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
10 eqeq2 2187 . . . . . . 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 4595 . . . . . . 7 ∅ ∈ ω
13 oveq2 5885 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
1413eqeq1d 2186 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +o 𝑥) = 𝐵 ↔ (𝐴 +o ∅) = 𝐵))
1514rspcev 2843 . . . . . . 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 6499 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1918biimp3a 1345 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
2019pm2.21d 619 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵𝐴 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
217, 17, 203jaod 1304 . . . 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 1205 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
24 nnaword1 6516 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑥))
25 sseq2 3181 . . . . 5 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
2624, 25syl5ibcom 155 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝐵𝐴𝐵))
2726rexlimdva 2594 . . 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 977  w3a 978   = wceq 1353  wcel 2148  wrex 2456  wss 3131  c0 3424  ωcom 4591  (class class class)co 5877   +o coa 6416
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423
This theorem is referenced by:  prarloclemn  7500
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