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Theorem oaun3lem2 42869
Description: The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 42876. (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 483 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → 𝑥 = (𝑎 +o 𝑏))
2 onelon 6389 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑎𝐴) → 𝑎 ∈ On)
32ad2ant2r 745 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎 ∈ On)
4 onelon 6389 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
54ad2ant2l 744 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑏 ∈ On)
6 oacl 8554 . . . . . . . . 9 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 𝑏) ∈ On)
73, 5, 6syl2anc 582 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 +o 𝑏) ∈ On)
8 oacl 8554 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
98adantr 479 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝐴 +o 𝐵) ∈ On)
107, 9jca 510 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → ((𝑎 +o 𝑏) ∈ On ∧ (𝐴 +o 𝐵) ∈ On))
11 simpl 481 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
1211adantr 479 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝐴 ∈ On)
133, 12, 53jca 1125 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 ∈ On ∧ 𝐴 ∈ On ∧ 𝑏 ∈ On))
14 simpl 481 . . . . . . . . . . . 12 ((𝑎𝐴𝑏𝐵) → 𝑎𝐴)
1514adantl 480 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
16 eloni 6374 . . . . . . . . . . . . . 14 (𝑎 ∈ On → Ord 𝑎)
17 eloni 6374 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
1816, 17anim12i 611 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝐴 ∈ On) → (Ord 𝑎 ∧ Ord 𝐴))
193, 12, 18syl2anc 582 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (Ord 𝑎 ∧ Ord 𝐴))
20 ordelpss 6392 . . . . . . . . . . . 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 4089 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
24 oawordri 8569 . . . . . . . . 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 pm3.22 458 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
2726adantr 479 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
28 simpr 483 . . . . . . . . . 10 ((𝑎𝐴𝑏𝐵) → 𝑏𝐵)
2928adantl 480 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑏𝐵)
30 oaordi 8565 . . . . . . . . 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 510 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → ((𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏) ∧ (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
33 ontr2 6411 . . . . . . 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 479 . . . . 5 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → (𝑎 +o 𝑏) ∈ (𝐴 +o 𝐵))
361, 35eqeltrd 2825 . . . 4 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → 𝑥 ∈ (𝐴 +o 𝐵))
3736exp31 418 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑎𝐴𝑏𝐵) → (𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵))))
3837rexlimdvv 3201 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑎𝐴𝑏𝐵 𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵)))
3938abssdv 4057 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 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2702  wrex 3060  wss 3939  wpss 3940  Ord word 6363  Oncon0 6364  (class class class)co 7416   +o coa 8482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  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 5144  df-opab 5206  df-mpt 5227  df-tr 5261  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-oadd 8489
This theorem is referenced by:  oaun3lem3  42870  oaun3lem4  42871  oaun3  42876
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