Theorem oaordex 8542

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 6404	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
2	1	adantl 486	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
3		oawordex 8541	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
4	2, 3	sylibd 242	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
5		oaord1 8535	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
6		eleq2 2858	. . . . . . . . . . . . 13 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ 𝐵))
7	5, 6	sylan9bb 518	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
8	7	biimprcd 253	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
9	8	exp4c 437	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → ∅ ∈ 𝑥))))
10	9	com12 33	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ∈ 𝐵 → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → ∅ ∈ 𝑥))))
11	10	imp4b 426	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12		simpr 489	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
13	11, 12	jca2 522	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
14	13	expd 420	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
15	14	reximdvai 3182	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
16	15	ex 417	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ 𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
17	16	adantr 485	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
18	4, 17	mpdd 44	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
19	7	biimpd 232	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 → 𝐴 ∈ 𝐵))
20	19	exp31 424	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 → 𝐴 ∈ 𝐵))))
21	20	com34 92	. . . . 5 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → (∅ ∈ 𝑥 → ((𝐴 +o 𝑥) = 𝐵 → 𝐴 ∈ 𝐵))))
22	21	imp4a 427	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)))
23	22	rexlimdv 3170	. . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
24	23	adantr 485	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
25	18, 24	impbid 215	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ⊆ wss 3913  ∅c0 4294  Oncon0 6361  (class class class)co 7411   +_o coa 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456

This theorem is referenced by: (None)