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Theorem oaun3lem2 43365
Description: The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 43372. (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 6411 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑎𝐴) → 𝑎 ∈ On)
32ad2ant2r 747 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎 ∈ On)
4 onelon 6411 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
54ad2ant2l 746 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑏 ∈ On)
6 oacl 8572 . . . . . . . . 9 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 𝑏) ∈ On)
73, 5, 6syl2anc 584 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 +o 𝑏) ∈ On)
8 oacl 8572 . . . . . . . . 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 1127 . . . . . . . . 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 6396 . . . . . . . . . . . . . 14 (𝑎 ∈ On → Ord 𝑎)
17 eloni 6396 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
1816, 17anim12i 613 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ 𝐴 ∈ On) → (Ord 𝑎 ∧ Ord 𝐴))
193, 12, 18syl2anc 584 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (Ord 𝑎 ∧ Ord 𝐴))
20 ordelpss 6414 . . . . . . . . . . . 12 ((Ord 𝑎 ∧ Ord 𝐴) → (𝑎𝐴𝑎𝐴))
2119, 20syl 17 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → (𝑎𝐴𝑎𝐴))
2215, 21mpbid 232 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
2322pssssd 4110 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
24 oawordri 8587 . . . . . . . . 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 8583 . . . . . . . . 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 6433 . . . . . . 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 2839 . . . 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 3210 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑎𝐴𝑏𝐵 𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵)))
3938abssdv 4078 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  wss 3963  wpss 3964  Ord word 6385  Oncon0 6386  (class class class)co 7431   +o coa 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509
This theorem is referenced by:  oaun3lem3  43366  oaun3lem4  43367  oaun3  43372
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