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Mirrors > Home > MPE Home > Th. List > enpr2d | Structured version Visualization version GIF version |
Description: A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-un 7677. (Revised by BTernaryTau, 23-Dec-2024.) |
Ref | Expression |
---|---|
enpr2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
enpr2d.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
enpr2d.3 | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
enpr2d | ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enpr2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | enpr2d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
3 | 0ex 5269 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
5 | 1oex 8427 | . . . 4 ⊢ 1o ∈ V | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) |
7 | enpr2d.3 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
8 | 7 | neqned 2951 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
9 | 1n0 8439 | . . . . 5 ⊢ 1o ≠ ∅ | |
10 | 9 | necomi 2999 | . . . 4 ⊢ ∅ ≠ 1o |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ≠ 1o) |
12 | 1, 2, 4, 6, 8, 11 | en2prd 8999 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {∅, 1o}) |
13 | df2o3 8425 | . 2 ⊢ 2o = {∅, 1o} | |
14 | 12, 13 | 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 Vcvv 3448 ∅c0 4287 {cpr 4593 class class class wbr 5110 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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 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: 1sdom2dom 9198 enpr2 9945 simpgnsgd 19886 2nsgsimpgd 19888 |
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