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Theorem oaordex 7882
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 5989 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
21adantl 469 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
3 oawordex 7881 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
42, 3sylibd 230 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
5 oaord1 7875 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
6 eleq2 2885 . . . . . . . . . . . . 13 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
75, 6sylan9bb 501 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (∅ ∈ 𝑥𝐴𝐵))
87biimprcd 241 . . . . . . . . . . 11 (𝐴𝐵 → (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∅ ∈ 𝑥))
98exp4c 421 . . . . . . . . . 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 410 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12 simpr 473 . . . . . . . . 9 ((𝑥 ∈ On ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (𝐴 +𝑜 𝑥) = 𝐵)
1312a1i 11 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (𝐴 +𝑜 𝑥) = 𝐵))
1411, 13jcad 504 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
1514expd 402 . . . . . 6 ((𝐴 ∈ On ∧ 𝐴𝐵) → (𝑥 ∈ On → ((𝐴 +𝑜 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
1615reximdvai 3213 . . . . 5 ((𝐴 ∈ On ∧ 𝐴𝐵) → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
1716ex 399 . . . 4 (𝐴 ∈ On → (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
1817adantr 468 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
194, 18mpdd 43 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
207biimpd 220 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (∅ ∈ 𝑥𝐴𝐵))
2120exp31 408 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +𝑜 𝑥) = 𝐵 → (∅ ∈ 𝑥𝐴𝐵))))
2221com34 91 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ On → (∅ ∈ 𝑥 → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))))
2322imp4a 411 . . . 4 (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵)))
2423rexlimdv 3229 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2524adantr 468 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2619, 25impbid 203 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wrex 3108  wss 3780  c0 4127  Oncon0 5947  (class class class)co 6881   +𝑜 coa 7800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-om 7303  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-oadd 7807
This theorem is referenced by: (None)
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