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Theorem nnaordex 7763
Description: Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaordex ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnaordex
StepHypRef Expression
1 nnon 7113 . . . . . 6 (𝐵 ∈ ω → 𝐵 ∈ On)
21adantl 481 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ On)
3 onelss 5804 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
42, 3syl 17 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
5 nnawordex 7762 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
64, 5sylibd 229 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
7 simplr 807 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → 𝐴𝐵)
8 eleq2 2719 . . . . . . . . 9 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
97, 8syl5ibrcom 237 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴 ∈ (𝐴 +𝑜 𝑥)))
10 peano1 7127 . . . . . . . . . . . 12 ∅ ∈ ω
11 nnaord 7744 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
1210, 11mp3an1 1451 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
1312ancoms 468 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
14 nna0 7729 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1514adantr 480 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 ∅) = 𝐴)
1615eleq1d 2715 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴 ∈ (𝐴 +𝑜 𝑥)))
1713, 16bitrd 268 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
1817adantlr 751 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
199, 18sylibrd 249 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵 → ∅ ∈ 𝑥))
2019ancrd 576 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
2120reximdva 3046 . . . . 5 ((𝐴 ∈ ω ∧ 𝐴𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
2221ex 449 . . . 4 (𝐴 ∈ ω → (𝐴𝐵 → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
2322adantr 480 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
246, 23mpdd 43 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
2517biimpa 500 . . . . . 6 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ ∅ ∈ 𝑥) → 𝐴 ∈ (𝐴 +𝑜 𝑥))
2625, 8syl5ibcom 235 . . . . 5 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ ∅ ∈ 𝑥) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2726expimpd 628 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2827rexlimdva 3060 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2928adantr 480 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
3024, 29impbid 202 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  wss 3607  c0 3948  Oncon0 5761  (class class class)co 6690  ωcom 7107   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609
This theorem is referenced by:  ltexpi  9762
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