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Mirrors > Home > MPE Home > Th. List > enpr2dOLD | Structured version Visualization version GIF version |
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.) |
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 9042 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → {𝐴} ≈ 1o) |
4 | enpr2dOLD.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
5 | 1on 8497 | . . . . 5 ⊢ 1o ∈ On | |
6 | en2sn 9064 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 1o ∈ On) → {𝐵} ≈ {1o}) | |
7 | 4, 5, 6 | sylancl 584 | . . . 4 ⊢ (𝜑 → {𝐵} ≈ {1o}) |
8 | enpr2dOLD.3 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
9 | 8 | neqned 2937 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | disjsn2 4712 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
12 | 5 | onirri 6477 | . . . . . 6 ⊢ ¬ 1o ∈ 1o |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → ¬ 1o ∈ 1o) |
14 | disjsn 4711 | . . . . 5 ⊢ ((1o ∩ {1o}) = ∅ ↔ ¬ 1o ∈ 1o) | |
15 | 13, 14 | sylibr 233 | . . . 4 ⊢ (𝜑 → (1o ∩ {1o}) = ∅) |
16 | unen 9069 | . . . 4 ⊢ ((({𝐴} ≈ 1o ∧ {𝐵} ≈ {1o}) ∧ (({𝐴} ∩ {𝐵}) = ∅ ∧ (1o ∩ {1o}) = ∅)) → ({𝐴} ∪ {𝐵}) ≈ (1o ∪ {1o})) | |
17 | 3, 7, 11, 15, 16 | syl22anc 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}) | |
20 | 17, 18, 19 | 3brtr4g 5177 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ suc 1o) |
21 | df-2o 8486 | . 2 ⊢ 2o = suc 1o | |
22 | 20, 21 | breqtrrdi 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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