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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 7718. (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 5297 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
5 | 1oex 8471 | . . . 4 ⊢ 1o ∈ V | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) |
7 | enpr2d.3 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
8 | 7 | neqned 2939 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
9 | 1n0 8483 | . . . . 5 ⊢ 1o ≠ ∅ | |
10 | 9 | necomi 2987 | . . . 4 ⊢ ∅ ≠ 1o |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ≠ 1o) |
12 | 1, 2, 4, 6, 8, 11 | en2prd 9044 | . 2 ⊢ (𝜑 → {𝐴, 𝐵} ≈ {∅, 1o}) |
13 | df2o3 8469 | . 2 ⊢ 2o = {∅, 1o} | |
14 | 12, 13 | breqtrrdi 5180 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∅c0 4314 {cpr 4622 class class class wbr 5138 1oc1o 8454 2oc2o 8455 ≈ cen 8932 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-suc 6360 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-1o 8461 df-2o 8462 df-en 8936 |
This theorem is referenced by: 1sdom2dom 9243 enpr2 9993 simpgnsgd 20012 2nsgsimpgd 20014 |
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