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Theorem oaordex 7623
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 5754 . . . . 5 (𝐵 ∈ On → (𝐴𝐵𝐴𝐵))
21adantl 482 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
3 oawordex 7622 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
42, 3sylibd 229 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
5 oaord1 7616 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +𝑜 𝑥)))
6 eleq2 2688 . . . . . . . . . . . . 13 ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴𝐵))
75, 6sylan9bb 735 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (∅ ∈ 𝑥𝐴𝐵))
87biimprcd 240 . . . . . . . . . . 11 (𝐴𝐵 → (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∅ ∈ 𝑥))
98exp4c 635 . . . . . . . . . 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 612 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +𝑜 𝑥) = 𝐵) → ∅ ∈ 𝑥))
12 simpr 477 . . . . . . . . 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 555 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐴𝐵) → ((𝑥 ∈ On ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
1514expd 452 . . . . . 6 ((𝐴 ∈ On ∧ 𝐴𝐵) → (𝑥 ∈ On → ((𝐴 +𝑜 𝑥) = 𝐵 → (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
1615reximdvai 3012 . . . . 5 ((𝐴 ∈ On ∧ 𝐴𝐵) → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
1716ex 450 . . . 4 (𝐴 ∈ On → (𝐴𝐵 → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
1817adantr 481 . . 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 219 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → (∅ ∈ 𝑥𝐴𝐵))
2120exp31 629 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ On → ((𝐴 +𝑜 𝑥) = 𝐵 → (∅ ∈ 𝑥𝐴𝐵))))
2221com34 91 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ On → (∅ ∈ 𝑥 → ((𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))))
2322imp4a 613 . . . 4 (𝐴 ∈ On → (𝑥 ∈ On → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵)))
2423rexlimdv 3026 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2524adantr 481 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵))
2619, 25impbid 202 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wrex 2910  wss 3567  c0 3907  Oncon0 5711  (class class class)co 6635   +𝑜 coa 7542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549
This theorem is referenced by: (None)
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