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Theorem nnawordex 6132
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 6103 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
213adant3 935 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3 nnaordex 6131 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
4 simpr 107 . . . . . . . 8 ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (𝐴 +𝑜 𝑥) = 𝐵)
54reximi 2433 . . . . . . 7 (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
63, 5syl6bi 156 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
763adant3 935 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
8 nna0 6084 . . . . . . . 8 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
983ad2ant1 936 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
10 eqeq2 2065 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 +𝑜 ∅) = 𝐴 ↔ (𝐴 +𝑜 ∅) = 𝐵))
119, 10syl5ibcom 148 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 = 𝐵 → (𝐴 +𝑜 ∅) = 𝐵))
12 peano1 4345 . . . . . . 7 ∅ ∈ ω
13 oveq2 5548 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
1413eqeq1d 2064 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 ∅) = 𝐵))
1514rspcev 2673 . . . . . . 7 ((∅ ∈ ω ∧ (𝐴 +𝑜 ∅) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
1612, 15mpan 408 . . . . . 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 6105 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
1918biimp3a 1251 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
2019pm2.21d 559 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵𝐴 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
217, 17, 203jaod 1210 . . . 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 1117 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
24 nnaword1 6117 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
25 sseq2 2995 . . . . 5 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
2624, 25syl5ibcom 148 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2726rexlimdva 2450 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2827adantr 265 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2923, 28impbid 124 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 101  wb 102  w3o 895  w3a 896   = wceq 1259  wcel 1409  wrex 2324  wss 2945  c0 3252  ωcom 4341  (class class class)co 5540   +𝑜 coa 6029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036
This theorem is referenced by:  prarloclemn  6655
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