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Theorem oaordex 8560
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) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oaordex
StepHypRef Expression
1 onelss 6405 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
21adantl 480 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
3 oawordex 8559 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
42, 3sylibd 238 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
5 oaord1 8553 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +o 𝑥)))
6 eleq2 2820 . . . . . . . . . . . . 13 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
75, 6sylan9bb 508 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥𝐴𝐵))
87biimprcd 249 . . . . . . . . . . 11 (𝐴𝐵 → (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
98exp4c 431 . . . . . . . . . 10 (𝐴𝐵 → (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → ∅ ∈ 𝑥))))
109com12 32 . . . . . . . . 9 (𝐴 ∈ On → (𝐴𝐵 → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → ∅ ∈ 𝑥))))
1110imp4b 420 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12 simpr 483 . . . . . . . 8 ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (𝐴 +o 𝑥) = 𝐵)
1311, 12jca2 512 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
1413expd 414 . . . . . 6 ((𝐴 ∈ On ∧ 𝐴𝐵) → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
1514reximdvai 3163 . . . . 5 ((𝐴 ∈ On ∧ 𝐴𝐵) → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
1615ex 411 . . . 4 (𝐴 ∈ On → (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
1716adantr 479 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
184, 17mpdd 43 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
197biimpd 228 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +o 𝑥) = 𝐵) → (∅ ∈ 𝑥𝐴𝐵))
2019exp31 418 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +o 𝑥) = 𝐵 → (∅ ∈ 𝑥𝐴𝐵))))
2120com34 91 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ On → (∅ ∈ 𝑥 → ((𝐴 +o 𝑥) = 𝐵𝐴𝐵))))
2221imp4a 421 . . . 4 (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵)))
2322rexlimdv 3151 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
2423adantr 479 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
2518, 24impbid 211 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wrex 3068  wss 3947  c0 4321  Oncon0 6363  (class class class)co 7411   +o coa 8465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-oadd 8472
This theorem is referenced by: (None)
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