Theorem enpr2dOLD 9095

Description: Obsolete version of enpr2d 9094 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	⊢ (𝜑 → 𝐴 ∈ 𝐶)
enpr2dOLD.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
enpr2dOLD.3	⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Assertion

Ref	Expression
enpr2dOLD	⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o)

Proof of Theorem enpr2dOLD

Step	Hyp	Ref	Expression
1		enpr2dOLD.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
2		ensn1g 9067	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o)
4		enpr2dOLD.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
5		1on 8523	. . . . 5 ⊢ 1o ∈ On
6		en2sn 9086	. . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o})
7	4, 5, 6	sylancl 586	. . . 4 ⊢ (𝜑 → {𝐵} ≈ {1o})
8		enpr2dOLD.3	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
9	8	neqned 2946	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
10		disjsn2 4718	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅)
12	5	onirri 6502	. . . . . 6 ⊢ ¬ 1o ∈ 1o
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o)
14		disjsn 4717	. . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o)
15	13, 14	sylibr 234	. . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅)
16		unen 9091	. . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o}))
17	3, 7, 11, 15, 16	syl22anc 839	. . 3 ⊢ (𝜑 → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o}))
18		df-pr 4635	. . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
19		df-suc 6395	. . 3 ⊢ suc 1o = (1o ∪ {1o})
20	17, 18, 19	3brtr4g 5183	. 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o)
21		df-2o 8512	. 2 ⊢ 2o = suc 1o
22	20, 21	breqtrrdi 5191	1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2107   ≠ wne 2939   ∪ cun 3962   ∩ cin 3963  ∅c0 4340  {csn 4632  {cpr 4634   class class class wbr 5149  Oncon0 6389  suc csuc 6391  1_oc1o 8504  2_oc2o 8505   ≈ cen 8987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-pss 3984  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-tr 5267  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5642  df-we 5644  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-ord 6392  df-on 6393  df-suc 6395  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-1o 8511  df-2o 8512  df-en 8991

This theorem is referenced by: (None)