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Theorem nnaordex 7703
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 7056 . . . . . 6 (𝐵 ∈ ω → 𝐵 ∈ On)
21adantl 482 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ On)
3 onelss 5754 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
42, 3syl 17 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
5 nnawordex 7702 . . . 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 791 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → 𝐴𝐵)
8 eleq2 2688 . . . . . . . . 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 7070 . . . . . . . . . . . 12 ∅ ∈ ω
11 nnaord 7684 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
1210, 11mp3an1 1409 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
1312ancoms 469 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
14 nna0 7669 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1514adantr 481 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 ∅) = 𝐴)
1615eleq1d 2684 . . . . . . . . . 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 750 . . . . . . . 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 3014 . . . . 5 ((𝐴 ∈ ω ∧ 𝐴𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
2221ex 450 . . . 4 (𝐴 ∈ ω → (𝐴𝐵 → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
2322adantr 481 . . 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 501 . . . . . 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 3027 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2928adantr 481 . 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 384   = wceq 1481  wcel 1988  wrex 2910  wss 3567  c0 3907  Oncon0 5711  (class class class)co 6635  ωcom 7050   +𝑜 coa 7542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549
This theorem is referenced by:  ltexpi  9709
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