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Theorem enpr2dOLD 9073
Description: Obsolete version of enpr2d 9072 as of 23-Dec-2024. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
enpr2dOLD.1 (𝜑𝐴𝐶)
enpr2dOLD.2 (𝜑𝐵𝐷)
enpr2dOLD.3 (𝜑 → ¬ 𝐴 = 𝐵)
Assertion
Ref Expression
enpr2dOLD (𝜑 → {𝐴, 𝐵} ≈ 2o)

Proof of Theorem enpr2dOLD
StepHypRef Expression
1 enpr2dOLD.1 . . . . 5 (𝜑𝐴𝐶)
2 ensn1g 9042 . . . . 5 (𝐴𝐶 → {𝐴} ≈ 1o)
31, 2syl 17 . . . 4 (𝜑 → {𝐴} ≈ 1o)
4 enpr2dOLD.2 . . . . 5 (𝜑𝐵𝐷)
5 1on 8497 . . . . 5 1o ∈ On
6 en2sn 9064 . . . . 5 ((𝐵𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o})
74, 5, 6sylancl 584 . . . 4 (𝜑 → {𝐵} ≈ {1o})
8 enpr2dOLD.3 . . . . . 6 (𝜑 → ¬ 𝐴 = 𝐵)
98neqned 2937 . . . . 5 (𝜑𝐴𝐵)
10 disjsn2 4712 . . . . 5 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
119, 10syl 17 . . . 4 (𝜑 → ({𝐴} ∩ {𝐵}) = ∅)
125onirri 6477 . . . . . 6 ¬ 1o ∈ 1o
1312a1i 11 . . . . 5 (𝜑 → ¬ 1o ∈ 1o)
14 disjsn 4711 . . . . 5 ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
1513, 14sylibr 233 . . . 4 (𝜑 → (1o ∩ {1o}) = ∅)
16 unen 9069 . . . 4 ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o}))
173, 7, 11, 15, 16syl22anc 837 . . 3 (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o}))
18 df-pr 4627 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
19 df-suc 6370 . . 3 suc 1o = (1o ∪ {1o})
2017, 18, 193brtr4g 5177 . 2 (𝜑 → {𝐴, 𝐵} ≈ suc 1o)
21 df-2o 8486 . 2 2o = suc 1o
2220, 21breqtrrdi 5185 1 (𝜑 → {𝐴, 𝐵} ≈ 2o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  wne 2930  cun 3937  cin 3938  c0 4318  {csn 4624  {cpr 4626   class class class wbr 5143  Oncon0 6364  suc csuc 6366  1oc1o 8478  2oc2o 8479  cen 8959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-ord 6367  df-on 6368  df-suc 6370  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-1o 8485  df-2o 8486  df-en 8963
This theorem is referenced by: (None)
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