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Theorem oaun3lem2 42717
Description: The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 42724. (Contributed by RP, 13-Feb-2025.)
Assertion
Ref Expression
oaun3lem2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎𝐴𝑏𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑥   𝐵,𝑎,𝑏,𝑥

Proof of Theorem oaun3lem2
StepHypRef Expression
1 simpr 484 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → 𝑥 = (𝑎 +o 𝑏))
2 onelon 6388 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑎𝐴) → 𝑎 ∈ On)
32ad2ant2r 746 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎 ∈ On)
4 onelon 6388 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
54ad2ant2l 745 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑏 ∈ On)
6 oacl 8547 . . . . . . . . 9 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 𝑏) ∈ On)
73, 5, 6syl2anc 583 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 +o 𝑏) ∈ On)
8 oacl 8547 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
98adantr 480 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝐴 +o 𝐵) ∈ On)
107, 9jca 511 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → ((𝑎 +o 𝑏) ∈ On ∧ (𝐴 +o 𝐵) ∈ On))
11 simpl 482 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
1211adantr 480 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝐴 ∈ On)
133, 12, 53jca 1126 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 ∈ On ∧ 𝐴 ∈ On ∧ 𝑏 ∈ On))
14 simpl 482 . . . . . . . . . . . 12 ((𝑎𝐴𝑏𝐵) → 𝑎𝐴)
1514adantl 481 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
16 eloni 6373 . . . . . . . . . . . . . 14 (𝑎 ∈ On → Ord 𝑎)
17 eloni 6373 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
1816, 17anim12i 612 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝐴 ∈ On) → (Ord 𝑎 ∧ Ord 𝐴))
193, 12, 18syl2anc 583 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (Ord 𝑎 ∧ Ord 𝐴))
20 ordelpss 6391 . . . . . . . . . . . 12 ((Ord 𝑎 ∧ Ord 𝐴) → (𝑎𝐴𝑎𝐴))
2119, 20syl 17 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎𝐴𝑎𝐴))
2215, 21mpbid 231 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
2322pssssd 4093 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
24 oawordri 8562 . . . . . . . . 9 ((𝑎 ∈ On ∧ 𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝑎𝐴 → (𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏)))
2513, 23, 24sylc 65 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏))
26 pm3.22 459 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
2726adantr 480 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
28 simpr 484 . . . . . . . . . 10 ((𝑎𝐴𝑏𝐵) → 𝑏𝐵)
2928adantl 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑏𝐵)
30 oaordi 8558 . . . . . . . . 9 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑏𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
3127, 29, 30sylc 65 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
3225, 31jca 511 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → ((𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏) ∧ (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
33 ontr2 6410 . . . . . . 7 (((𝑎 +o 𝑏) ∈ On ∧ (𝐴 +o 𝐵) ∈ On) → (((𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏) ∧ (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)) → (𝑎 +o 𝑏) ∈ (𝐴 +o 𝐵)))
3410, 32, 33sylc 65 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 +o 𝑏) ∈ (𝐴 +o 𝐵))
3534adantr 480 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → (𝑎 +o 𝑏) ∈ (𝐴 +o 𝐵))
361, 35eqeltrd 2828 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → 𝑥 ∈ (𝐴 +o 𝐵))
3736exp31 419 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑎𝐴𝑏𝐵) → (𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵))))
3837rexlimdvv 3205 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑎𝐴𝑏𝐵 𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵)))
3938abssdv 4061 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎𝐴𝑏𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  {cab 2704  wrex 3065  wss 3944  wpss 3945  Ord word 6362  Oncon0 6363  (class class class)co 7414   +o coa 8475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-oadd 8482
This theorem is referenced by:  oaun3lem3  42718  oaun3lem4  42719  oaun3  42724
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