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Mirrors > Home > MPE Home > Th. List > enpr2dOLD | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
enpr2dOLD.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
enpr2dOLD.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
enpr2dOLD.3 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
enpr2dOLD | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enpr2dOLD.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | ensn1g 8970 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o) |
4 | enpr2dOLD.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
5 | 1on 8429 | . . . . 5 ⊢ 1o ∈ On | |
6 | en2sn 8992 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o}) | |
7 | 4, 5, 6 | sylancl 587 | . . . 4 ⊢ (𝜑 → {𝐵} ≈ {1o}) |
8 | enpr2dOLD.3 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
9 | 8 | neqned 2951 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | disjsn2 4678 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
12 | 5 | onirri 6435 | . . . . . 6 ⊢ ¬ 1o ∈ 1o |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o) |
14 | disjsn 4677 | . . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o) | |
15 | 13, 14 | sylibr 233 | . . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅) |
16 | unen 8997 | . . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) | |
17 | 3, 7, 11, 15, 16 | syl22anc 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}) | |
20 | 17, 18, 19 | 3brtr4g 5144 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o) |
21 | df-2o 8418 | . 2 ⊢ 2o = suc 1o | |
22 | 20, 21 | breqtrrdi 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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