Theorem oaordex 8483

Description: Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse. (Contributed by NM, 12-Dec-2004.)

Assertion

Ref	Expression
oaordex	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oaordex

Step	Hyp	Ref	Expression
1		onelss 6357	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
2	1	adantl 481	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
3		oawordex 8482	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
4	2, 3	sylibd 239	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
5		oaord1 8476	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
6		eleq2 2823	. . . . . . . . . . . . 13 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ 𝐵))
7	5, 6	sylan9bb 509	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
8	7	biimprcd 250	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
9	8	exp4c 432	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → ∅ ∈ 𝑥))))
10	9	com12 32	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ∈ 𝐵 → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → ∅ ∈ 𝑥))))
11	10	imp4b 421	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
13	11, 12	jca2 513	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
14	13	expd 415	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
15	14	reximdvai 3145	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
16	15	ex 412	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ 𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
17	16	adantr 480	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
18	4, 17	mpdd 43	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
19	7	biimpd 229	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 → 𝐴 ∈ 𝐵))
20	19	exp31 419	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 → 𝐴 ∈ 𝐵))))
21	20	com34 91	. . . . 5 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → (∅ ∈ 𝑥 → ((𝐴 +o 𝑥) = 𝐵 → 𝐴 ∈ 𝐵))))
22	21	imp4a 422	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)))
23	22	rexlimdv 3133	. . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
24	23	adantr 480	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
25	18, 24	impbid 212	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ⊆ wss 3899  ∅c0 4283  Oncon0 6315  (class class class)co 7356   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399

This theorem is referenced by: (None)