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Theorem oaun3lem2 42701
Description: The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 42708. (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 6383 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑎𝐴) → 𝑎 ∈ On)
32ad2ant2r 744 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎 ∈ On)
4 onelon 6383 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑏𝐵) → 𝑏 ∈ On)
54ad2ant2l 743 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑏 ∈ On)
6 oacl 8536 . . . . . . . . 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 8536 . . . . . . . . 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 1125 . . . . . . . . 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 6368 . . . . . . . . . . . . . 14 (𝑎 ∈ On → Ord 𝑎)
17 eloni 6368 . . . . . . . . . . . . . 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 6386 . . . . . . . . . . . 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 4092 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎𝐴𝑏𝐵)) → 𝑎𝐴)
24 oawordri 8551 . . . . . . . . 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 8547 . . . . . . . . 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 6405 . . . . . . 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 2827 . . . 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 3204 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑎𝐴𝑏𝐵 𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵)))
3938abssdv 4060 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 1084   = wceq 1533  wcel 2098  {cab 2703  wrex 3064  wss 3943  wpss 3944  Ord word 6357  Oncon0 6358  (class class class)co 7405   +o coa 8464
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-oadd 8471
This theorem is referenced by:  oaun3lem3  42702  oaun3lem4  42703  oaun3  42708
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