Theorem oaun3lem2 43358

Description: The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 43365. (Contributed by RP, 13-Feb-2025.)

Assertion

Ref	Expression
oaun3lem2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑥   𝐵,𝑎,𝑏,𝑥

Proof of Theorem oaun3lem2

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → 𝑥 = (𝑎 +o 𝑏))
2		onelon 6332	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ On)
3	2	ad2ant2r 747	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ On)
4		onelon 6332	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
5	4	ad2ant2l 746	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ On)
6		oacl 8453	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +o 𝑏) ∈ On)
7	3, 5, 6	syl2anc 584	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 +o 𝑏) ∈ On)
8		oacl 8453	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
9	8	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝐴 +o 𝐵) ∈ On)
10	7, 9	jca 511	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((𝑎 +o 𝑏) ∈ On ∧ (𝐴 +o 𝐵) ∈ On))
11		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
12	11	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐴 ∈ On)
13	3, 12, 5	3jca 1128	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 ∈ On ∧ 𝐴 ∈ On ∧ 𝑏 ∈ On))
14		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
15	14	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐴)
16		eloni 6317	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ On → Ord 𝑎)
17		eloni 6317	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → Ord 𝐴)
18	16, 17	anim12i 613	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝐴 ∈ On) → (Ord 𝑎 ∧ Ord 𝐴))
19	3, 12, 18	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (Ord 𝑎 ∧ Ord 𝐴))
20		ordelpss 6335	. . . . . . . . . . . 12 ⊢ ((Ord 𝑎 ∧ Ord 𝐴) → (𝑎 ∈ 𝐴 ↔ 𝑎 ⊊ 𝐴))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 ∈ 𝐴 ↔ 𝑎 ⊊ 𝐴))
22	15, 21	mpbid 232	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ⊊ 𝐴)
23	22	pssssd 4051	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ⊆ 𝐴)
24		oawordri 8468	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ 𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝑎 ⊆ 𝐴 → (𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏)))
25	13, 23, 24	sylc 65	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏))
26		pm3.22 459	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
28		simpr 484	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
29	28	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
30		oaordi 8464	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
31	27, 29, 30	sylc 65	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵))
32	25, 31	jca 511	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → ((𝑎 +o 𝑏) ⊆ (𝐴 +o 𝑏) ∧ (𝐴 +o 𝑏) ∈ (𝐴 +o 𝐵)))
33		ontr2 6355	. . . . . . 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)))
34	10, 32, 33	sylc 65	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 +o 𝑏) ∈ (𝐴 +o 𝐵))
35	34	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → (𝑎 +o 𝑏) ∈ (𝐴 +o 𝐵))
36	1, 35	eqeltrd 2828	. . . 4 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑥 = (𝑎 +o 𝑏)) → 𝑥 ∈ (𝐴 +o 𝐵))
37	36	exp31 419	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵))))
38	37	rexlimdvv 3185	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏) → 𝑥 ∈ (𝐴 +o 𝐵)))
39	38	abssdv 4020	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3903   ⊊ wpss 3904  Ord word 6306  Oncon0 6307  (class class class)co 7349   +_o coa 8385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-oadd 8392

This theorem is referenced by:  oaun3lem3  43359  oaun3lem4  43360  oaun3  43365