Theorem oaordex 8264

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 6233	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
2	1	adantl 485	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
3		oawordex 8263	. . . 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 8257	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
6		eleq2 2819	. . . . . . . . . . . . 13 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ 𝐵))
7	5, 6	sylan9bb 513	. . . . . . . . . . . 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 436	. . . . . . . . . 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 425	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12		simpr 488	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
13	11, 12	jca2 517	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
14	13	expd 419	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
15	14	reximdvai 3181	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
16	15	ex 416	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ 𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
17	16	adantr 484	. . 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 232	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 → 𝐴 ∈ 𝐵))
20	19	exp31 423	. . . . . 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 426	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)))
23	22	rexlimdv 3192	. . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
24	23	adantr 484	. 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 399   = wceq 1543   ∈ wcel 2112  ∃wrex 3052   ⊆ wss 3853  ∅c0 4223  Oncon0 6191  (class class class)co 7191   +_o coa 8177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-oadd 8184

This theorem is referenced by: (None)