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Theorem nnaordex 7955
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 7301 . . . . . 6 (𝐵 ∈ ω → 𝐵 ∈ On)
21adantl 469 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ∈ On)
3 onelss 5979 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
42, 3syl 17 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
5 nnawordex 7954 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
64, 5sylibd 230 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
7 simplr 776 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → 𝐴𝐵)
8 eleq2 2874 . . . . . . . . 9 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
97, 8syl5ibrcom 238 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴 ∈ (𝐴 +𝑜 𝑥)))
10 peano1 7315 . . . . . . . . . . . 12 ∅ ∈ ω
11 nnaord 7936 . . . . . . . . . . . 12 ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
1210, 11mp3an1 1565 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
1312ancoms 448 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
14 nna0 7921 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1514adantr 468 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 ∅) = 𝐴)
1615eleq1d 2870 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴 ∈ (𝐴 +𝑜 𝑥)))
1713, 16bitrd 270 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
1817adantlr 697 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
199, 18sylibrd 250 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵 → ∅ ∈ 𝑥))
2019ancrd 543 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐴𝐵) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
2120reximdva 3204 . . . . 5 ((𝐴 ∈ ω ∧ 𝐴𝐵) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
2221ex 399 . . . 4 (𝐴 ∈ ω → (𝐴𝐵 → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
2322adantr 468 . . 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 464 . . . . . 6 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ ∅ ∈ 𝑥) → 𝐴 ∈ (𝐴 +𝑜 𝑥))
2625, 8syl5ibcom 236 . . . . 5 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ ∅ ∈ 𝑥) → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
2726expimpd 443 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2827rexlimdva 3219 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2928adantr 468 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
3024, 29impbid 203 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wrex 3097  wss 3769  c0 4116  Oncon0 5936  (class class class)co 6874  ωcom 7295   +𝑜 coa 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-oadd 7800
This theorem is referenced by:  ltexpi  10009
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