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Theorem enpr2dOLD 9001
Description: Obsolete version of enpr2d 9000 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 8970 . . . . 5 (𝐴𝐶 → {𝐴} ≈ 1o)
31, 2syl 17 . . . 4 (𝜑 → {𝐴} ≈ 1o)
4 enpr2dOLD.2 . . . . 5 (𝜑𝐵𝐷)
5 1on 8429 . . . . 5 1o ∈ On
6 en2sn 8992 . . . . 5 ((𝐵𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o})
74, 5, 6sylancl 587 . . . 4 (𝜑 → {𝐵} ≈ {1o})
8 enpr2dOLD.3 . . . . . 6 (𝜑 → ¬ 𝐴 = 𝐵)
98neqned 2951 . . . . 5 (𝜑𝐴𝐵)
10 disjsn2 4678 . . . . 5 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
119, 10syl 17 . . . 4 (𝜑 → ({𝐴} ∩ {𝐵}) = ∅)
125onirri 6435 . . . . . 6 ¬ 1o ∈ 1o
1312a1i 11 . . . . 5 (𝜑 → ¬ 1o ∈ 1o)
14 disjsn 4677 . . . . 5 ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
1513, 14sylibr 233 . . . 4 (𝜑 → (1o ∩ {1o}) = ∅)
16 unen 8997 . . . 4 ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o}))
173, 7, 11, 15, 16syl22anc 838 . . 3 (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o}))
18 df-pr 4594 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
19 df-suc 6328 . . 3 suc 1o = (1o ∪ {1o})
2017, 18, 193brtr4g 5144 . 2 (𝜑 → {𝐴, 𝐵} ≈ suc 1o)
21 df-2o 8418 . 2 2o = suc 1o
2220, 21breqtrrdi 5152 1 (𝜑 → {𝐴, 𝐵} ≈ 2o)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wne 2944  cun 3913  cin 3914  c0 4287  {csn 4591  {cpr 4593   class class class wbr 5110  Oncon0 6322  suc csuc 6324  1oc1o 8410  2oc2o 8411  cen 8887
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-ord 6325  df-on 6326  df-suc 6328  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-1o 8417  df-2o 8418  df-en 8891
This theorem is referenced by: (None)
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