Theorem oaordex 8604

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 6431	. . . . 5 ⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
2	1	adantl 487	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
3		oawordex 8603	. . . 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 8597	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
6		eleq2 2853	. . . . . . . . . . . . 13 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ 𝐵))
7	5, 6	sylan9bb 519	. . . . . . . . . . . 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 438	. . . . . . . . . 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 427	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12		simpr 490	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
13	11, 12	jca2 523	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
14	13	expd 421	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
15	14	reximdvai 3177	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 ∈ 𝐵) → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
16	15	ex 418	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ 𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
17	16	adantr 486	. . 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 425	. . . . . 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 428	. . . 4 ⊢ (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)))
23	22	rexlimdv 3165	. . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
24	23	adantr 486	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  Oncon0 6388  (class class class)co 7451   +_o coa 8511

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-oadd 8518

This theorem is used by: (None)