Theorem oawordeu 8198

Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)

Assertion

Ref	Expression
oawordeu	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oawordeu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3920	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵))
2		oveq1 7164	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +o 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥))
3	2	eqeq1d 2761	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
4	3	reubidv 3308	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 ↔ ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
5	1, 4	imbi12d 348	. . 3 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) ↔ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵)))
6		sseq2 3921	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅)))
7		eqeq2 2771	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → ((if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = if(𝐵 ∈ On, 𝐵, ∅)))
8	7	reubidv 3308	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵 ↔ ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = if(𝐵 ∈ On, 𝐵, ∅)))
9	6, 8	imbi12d 348	. . 3 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → ((if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵) ↔ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅) → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = if(𝐵 ∈ On, 𝐵, ∅))))
10		0elon 6228	. . . . 5 ⊢ ∅ ∈ On
11	10	elimel 4493	. . . 4 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
12	10	elimel 4493	. . . 4 ⊢ if(𝐵 ∈ On, 𝐵, ∅) ∈ On
13		eqid 2759	. . . 4 ⊢ {𝑦 ∈ On ∣ if(𝐵 ∈ On, 𝐵, ∅) ⊆ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑦)} = {𝑦 ∈ On ∣ if(𝐵 ∈ On, 𝐵, ∅) ⊆ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑦)}
14	11, 12, 13	oawordeulem 8197	. . 3 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅) → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = if(𝐵 ∈ On, 𝐵, ∅))
15	5, 9, 14	dedth2h 4483	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
16	15	imp 410	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1539   ∈ wcel 2112  ∃!wreu 3073  {crab 3075   ⊆ wss 3861  ∅c0 4228  ifcif 4424  Oncon0 6175  (class class class)co 7157   +_o coa 8116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-oadd 8123

This theorem is referenced by:  oawordex  8200  oaf1o  8206  oaabs  8288  oaabs2  8289  finxpreclem4  35127