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Theorem nnsseleq 6302
Description: For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
Assertion
Ref Expression
nnsseleq ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Proof of Theorem nnsseleq
StepHypRef Expression
1 nntri1 6297 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
2 nntri3or 6294 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
3 df-3or 928 . . . . . 6 ((𝐴𝐵𝐴 = 𝐵𝐵𝐴) ↔ ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
42, 3sylib 121 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵𝐴 = 𝐵) ∨ 𝐵𝐴))
54orcomd 686 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵𝐴 ∨ (𝐴𝐵𝐴 = 𝐵)))
65ord 681 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ 𝐵𝐴 → (𝐴𝐵𝐴 = 𝐵)))
71, 6sylbid 149 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → (𝐴𝐵𝐴 = 𝐵)))
8 nnord 4454 . . . . 5 (𝐵 ∈ ω → Ord 𝐵)
98adantl 272 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → Ord 𝐵)
10 ordelss 4230 . . . . 5 ((Ord 𝐵𝐴𝐵) → 𝐴𝐵)
1110ex 114 . . . 4 (Ord 𝐵 → (𝐴𝐵𝐴𝐵))
129, 11syl 14 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
13 eqimss 3093 . . . 4 (𝐴 = 𝐵𝐴𝐵)
1413a1i 9 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
1512, 14jaod 675 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵𝐴 = 𝐵) → 𝐴𝐵))
167, 15impbid 128 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 667  w3o 926   = wceq 1296  wcel 1445  wss 3013  Ord word 4213  ωcom 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434
This theorem is referenced by:  frec2uzled  9985
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